Fine. Since I've no desire to talk politics here, I'll go ahead and address the question. Keep in mind that I've no interest is discussing specific real world examples, and I'm going to try to as much as possible avoid specific real world theories - although that's going to be somewhat impossible. So to get that out of the way I'll just say that the D&D alignment terms are descriptive, and not proscriptive. Each alignment and its proponents - including evil - believes it is in some sense 'right' and 'correct' and that it promotes 'right' and 'correct' behavior. There are real world examples of people who promote concepts normally thought of as evil as if they were good, or who declaim against concepts normally thought of as good as being in fact wrong. Since an alignment system requires us to take an absolute view, we have to sort those concepts into buckets based on their features, but in doing so we are - unlike what is accused by some - not taking a stand on who is right. As far as the alignment system is concerned, perhaps the Chaotic Neutrals who promote absolute rational self-interest and oppose altruism as a tool of subjugation are correct and the best possible approach to the reality of the world is Chaotic Neutral. We only label them Chaotic Neutral because it is clear that the heterodox and surprising idea that altruism is bad, while it may not be wrong, doesn't easily fit with what is normally thought of as 'Good' - even if it may turn out that 'Good' produces less good ends in practice than what we are calling CN.
This also shows why on a purely practical basis, it's a stupid idea to try to draw real world examples. Because the more real world examples we use, the more our own personal biases are going to color the results and the more we are going to end up arguing over how the bucket containing thing we prefer or self-identify with ought to be labeled.
That out of the way, hopefully their won't be too many arguments from the relative perspective that everyone's favorite ideology ought to the be one labeled 'Good' and we can dispense with useless assertions that morality is all relative anyway so the whole idea of an alignment system is wrong. (These assertions can often be labeled as aligned viewpoint assertions anyway, since in fact the Chaotic end of the spectrum largely agrees that morality is relative, although what they mean by that differs between CG and CE obviously. And obviously one way to describe the Neutral viewpoint is that morality is entirely an artificial construct. But back on topic.)
The first thing I'd like to note is that you have it backwards to a certain extent. The question isn't so much "How do Governments Align" but rather "How do Alignments Govern". Governments are manifestations of the underlying cultures that produce them, and while there is a bit of a feedback loop going on here so that the government in turn alters the culture, the primacy is rightly with the culture and not the government. Cultures have an almost unlimited capacity to change the government, but the government has only limited capacity to change the culture. It's a manifestation of what people believe, and its I think rightly true that "people get the government that they deserve, not the government they want".
And in that we get the first sense that its going to be messy. If we start talking about real governments and trying to align them, what we will notice is that they rarely if ever achieve a platonic idea of government. Instead, the government might be instituted under one set of principles, only to fail because the culture doesn't actually support them and be actually governed under an entirely different set. Or a government might deliberately be instituted under a lie, so that the face that the government wants to present of itself is entirely different than what it really is - the USSR's constitution famously on paper protected civil liberties even more than the American constitution, even though in practice real freedom of speech and freedom of the press didn't actually exist. Finally, and this is most common, a government might be instituted under one set of principles, only to have the culture shift out from under it somewhat, so that the resulting system becomes hybrid of what the older culture and those that cling to it intended and what the newer culture and its proponents would have created on their own. It's particularly common, and even predictable, that a Lawful culture which values Tradition, will have vestiges of irrational things that don't seem to make sense and which support a completely different set of notions simply because it doesn't like to abandon older rituals and rather fears rapid change.
Again, it should be obvious that I think all real world examples are very messy indeed, so when I start outlining idealized forms it will be a mistake to think things like, "Oh, that's the American system.", or think that if a description seems familiar that I'm labeling a real world nation by a particular label. Even where the shoe fits, the real system is going to be so messy and often evolving that if I actually attempted to describe a real nation (which I wouldn't, as it would be a rather useless exercise compared to creating fantasy nations to play in), it would be longer in some cases than this already long essay.
To make the most sense of this, we need to start by thinking about the sort of government that each set of ideas would create in its pure form, before we start dealing with the complexities of compromise between several cultures, cultural drift over time, and so forth.
You do seem to start with the proper basis, roughly getting your labels "Good", "Law", "Chaos", and "Evil" correctly described - or at least close enough for these purposes. So I'm going to avoid spending a lot of time describing the labels and assume that our ideas about what those words mean are close enough for a working start. I do want to caution you however, that it's actually the "at any cost" that is more important to the idea of evil than "self-interest" is. It's actually what it doesn't value - life, happiness, growth, health, etc. - that is more defining here than the fact that it values self. Indeed, there are forms of evil that don't value the self at all, but instead willingly sacrifice the self "at any cost" to serve the evil end. Likewise, while evil prides often itself on being "pragmatic", that really is only for certain values of "pragmatic". Evil values power over weal, for example, and force and the ability to enforce its desires over health. So for example, evil admires the person who in seeking vengeance destroys everything that they love and makes themselves into a heartless weapon stabbing at their foe even from the mouth of hell's pit in order to avenge themselves. Or evil holds that the narcotic addict stoned out of their mind lying in their filth is doing well in seeking their gratification above everything else. But one could I think argue that all those states are hardly practical for other values of practical.
Lawful Good: In ideal, a constitutional monarchy where the monarch himself is merely the embodiment and instrument of a higher set of laws which bind even him. Each person has a place in this structure and a purpose and a set of mutually ennobling relationships with peers, and those both higher and lower in the hierarchy, which is called duty. The King rules through his loyal vassals, who form a peerage whom he loves and delights to honor, and who in turn delight in doing the King's will. And this relationship extends downward at every level to the least member of society for whom the whole society actually exists and supports, so that the scullion girl and the maid who tends the fireplace is the friend and beloved of both her peers and her betters and each person finds themselves loved and supported and knows that they each in some small way contribute to the health of everyone in ways that are appreciated and honorable. Laws will tend to be numerous and harsh, but the society - being also good - will tend to mitigate this by the recognition that the cure for petty crimes is often worse than the ill that the propose to cure. So society will tend to favor enforcing most of its disapproval for smaller vices through social disapproval rather than judicial writ, leaving it up to the individuals to reform the situation before it becomes a greater vice. Thus, there will often be a measure of personal freedom and discretion, and in particular each lord will be encouraged to as much as possible take a merciful tact and apply the law in spirit rather than in impersonal letter, taking into account the desire of and the desirability of reforming everyone because - being Good - the society tends to err on the side of seeing everyone as having worth. As much as possible, the law will try to provide just and merciful institutions for handling the situation when thing go wrong, so that poverty, abandonment, and even crime are dealt with, but so that these institutions don't overly burden the whole of society by and large the expectation is that problems will be dealt with at below the level of institutional justice, with social enforcement. A family will for instance take care of its own as much as possible, not merely out of proper familial love, but out of a feeling of not wanting to burden others who already no doubt have their own cares. That is to say, they will do their duty toward that person.
Lawful Neutral: Much as Lawful Good, but without the (to its mind) foolish concern for mercy and personal freedom which just allows individuals to shirk their duties, and without as much concern for individual weal. The idea that you ought to serve your master because you love him and he loves you will be missing. Instead, you serve your master because he is your master, and the burden of responsibility will be more decidedly on the servant. The master has concern for things higher than the servant, which is the health of the whole, and so the servant must understand that if it seems he is ill-used it is still his privilege to serve. In all things, right order and duty must prevail. There will be little if any desire to attempt to reform the situation if things go wrong. Servants (or masters) that fail in their duty do not deserve consideration, and quite often the best they can hope for is to be allowed to destroy themselves to spare their betters the trouble of having to do it - that is, an honorable death. The tendency will be more toward Absolute Monarchy, with fewer safeguards on the exercise of legal authority, and the word of the Ruler being absolute. That said, there will be little expectation that anyone in the hierarchy has much room for exercising personal judgment, as the law and tradition will make each person's duty clear and inescapable. The idea is for the whole society to work like a machine which is instructed perfectly by the perfect law, however it comes down to them.
Lawful Evil: Much as lawful neutral, but the law will not be written merely with the idea that ideal society is a perfectly functioning machine, but that a perfectly functioning society is one that is unreservedly cruel and uncompromising. The Lawful Evil society doesn't naively expect its members to do their duty because it is their duty, but that each cog in the system must be compelled by force to do its duty and given no choice in the matter. The system works best when each servant suffers under the weight of the lash, obeys without hesitation because it is compelled by fear and pain, and has so lost its sense of identity that they cannot even conceive of rebelling. The society takes to heart the maxim that is it is better to be feared than loved. Any problem in the system, any crime against the society - and everything that is not 100% commitment to the society is a crime - must be not only rooted out, but punished disproportionately to the offense both to set an example and to properly allow the lord to avenge themselves against the hateful crime of being slighted in the least. Each person is placed in the role they are needed in and suited for without slightest regard for affection. Everything is a ruthless meritocracy. The tendency here will be toward Absolute Despotism. The lord has no preference for seeing his son succeed him. His chosen heir will be whoever is strongest, most ruthless, and most capable. A word of caution here though. Too often this system is ultimately described in terms that descend into Chaos with all sorts of backstabbing and treachery. It's important to understand that the vassals in this system, have no desire to improve their rank within the system save through the proper legal channels. No one in this system 'delights' to serve, but they do all desire to serve. And while the entire system exists to support the will of the absolute monarch, even he considers himself merely the most valuable most important cog in the machine, and if it ever comes to it willingly sacrifices himself for the sake of the whole. While the society collectively is selfish in that it desires to subjugate all other societies and assimilate them to itself, no individual component of the system is driven by self-interest. A good fictional example of this idealized state of affairs is in 1984, where 'Big Brother' doesn't even actually exist, but everyone is a cog that serves the system anyway even though no one truly enjoys doing so. The perfectly functioning Lawful Evil society is ultimately ruled by the social construct and doesn't even need a real leader at the top of it issuing commands. It just keeps going because there isn't anyway to evade it's continued functioning - everyone is under compulsion and even its theoretically absolute ruler couldn't alter that even if he impossibly wanted to.
Neutral Evil: For me, it's almost impossible to imagine a society which has become predominately neutral evil and persist in that state for very long. You see examples of it over short periods, such as the paroxysm of Nazi Germany in its final days where SS guards who knew the war was lost and as such everything they believed in was a lie, still persisted in trying to execute as many prisoners as possible, before committing suicide together with their families, or in the killing fields of Cambodia where the whole society seemed to be trying to destroy itself in a rush to see which group would be last to destroy the other. Even the existence of an evil society or any other sort of prosperity is antithetical to the goals of Neutral Evil, which ultimately is insisting that nothing is redeeming or worthwhile. But as there are short term real world examples, in a fantasy setting you can come up with examples that represent society in a perpetual dystopian state - such as a society where all the members are converted to undead or where the society is effectively ruled by some sort of destructive curse. In practice, most societies that you'd describe as predominately Neutral Evil are actually societies that are eclectically a mix of Chaotic Evil and Lawful Evil competing narratives warring with themselves in such a way that each fails to realize the result is in neither groups interest, but the paroxysm of annihilation is temporarily abated by each groups continuing to value one thing or another.
Chaotic Evil: Anarchy. In a certain sense, there is little else to say here, though I should note that here I mean Anarchy and not various Anarchy inspired systems of government. The perfect end goal of CE is not only no order, but no basis for relationships or social ideas that could interfere with the strong taking what they want from the weak. There is no government. No rules. In the real world, this happens in areas we call now 'failed states'. No one is concerned with much anything but survival. The largest sort of social organization observed is an ad hoc gang ruled by the best and most efficient bully for as long as he can hold the job. One word of caution though is that quite often, societies presented in D&D in the past as nearly pure 'chaotic evil' are nothing like this but instead fascist absolute monarchies ruled by complex traditions and manifesting all sorts of social institutions. It should be obvious that such societies are actually predominately Lawful Evil with at least some self-centered Chaotic Evil members or otherwise complex kludged examples and not nearly their platonic form. In my own mind, the purest Chaotic Evil social expression is real world leonine society, and my own CE societies (Drow, for instance) are actually closely modeled after the culture of lion prides.
Chaotic Neutral: Libertarian Democracy, although in the purest form, we ought to say government per se doesn't exist. Also all of those idealized Anarchy systems of government, which have the rather chaotic nature of being things that can't actually be systematically put into practice, but if you were to try to put them into practice and they weren't immediately subverted out of a misplaced desire to make them work, would be chaotic neutral. The point being however that the basis of society will be that in so far as you don't deprive others of their rights, you can and ought to be allowed to pretty well do what you please with at most the expectations that other people will show their individual displeasure by refusing to associate with you. The basis of all relationships is ultimately private and contractual, with authority only existing to ensure that those contracts are enforced without descending into total anarchy, freely entered into, and otherwise not oppressive over the long term and have a suitable means of ending the relationship equitably. But 'buyer beware' will tend to be the rule of the day and what authority there is will take a dim view of forcing you to observe rights you don't yourself protect or otherwise doing things for your own good, which you ought to know better than they do. Trial when it occurs will tend to be by a group of peers, and there will be a tendency toward government through ad hoc appointed committee with no lasting institutions. In practice, most things will be rather spontaneous by design. Militias will dominate over standing armies. Elected sheriffs will appoint citizens for special purpose law enforcement as and if it comes up. Elected judges will largely be responsible not for enforcing or interpreting law, but overseeing juries on a need basis who will do the enforcing and ruling and interpreting. Surprisingly perhaps, feudalism of a sort works pretty well here, provided we limit ourselves solely to the feudal system and assume that the institution of inheritable serfdom doesn't exist. Feudalism in the idea as well as the practice tended toward government through private contract and if you outlaw slavery and get rid of ideas like the divine right of kings, you end up with something rather like libertarian democracy - and its to be noted that this isn't a coincidence, but rather pretty much exactly where we got democracy from in the real world.
Chaotic Good: Democracy, often implemented as a Federal Republic or similar institution of distributed powers. Quite similar in many ways to Chaotic Neutral, but there will be some concessions to ensuring that the strong do not end up oppressing the weak, and that critically the rights of minorities in the system are protected from the power of the ad hoc mob. The law will primarily be organized on an individual rights and liberties basis, and typically the government will be strongly limited in its powers. There will be a strong distrust of centralized authority, and a strong tendency to delegate authority to local governments and indeed to individuals when possible. The law will typically be seen as flexible and evolving to meet changing needs. There will tend to be a preference for common law rather than written law, and interpreting each situation based on its particular facts. Political parties will tend to be numerous, and ever changing, as different associations and interests change over time, and the typical basis of both government in its ideal and practical form will be seen as friendship and personal relationships. The government governs best when it stays out of the way, but when it does intervene the magistrate is someone you know and indeed who has the position because you trusted him personally to do that job better than anyone else you knew.
Neutral Good: Parliamentary Republic or perhaps Syndical Oligarchy. Obviously, in many ways halfway between the Lawful Good description and the Chaotic Good description and with some features of each. I'm going to be deliberately vague here because it's important to understand that a true Neutral Good society doesn't actually consider the exact structure of the government all that important, as it is actually neutral on the question of whether order or the lack of it is what promotes weal. A true Neutral Good society would say that an over interest in the particulars of government rather missed the point, as the goal wasn't to create a government per se but to create weal and health and contentment by whatever tool was available and that government alone - nor its lack - couldn't accomplish or provide for that. Compared to Lawful Good, there will be a general distrust of a central decision maker, and committees of advisors or a parliament or a council of representatives (elected or otherwise) will be preferred over a single dictator and tend to hold the real power. Compared to Chaotic Good, there will be a general distrust that the body politic can be completely self-regulating and establish justice for all on its own, but conversely there will be far more strong guarantees of individual rights compared to Lawful Good and a general agreement that duty ought to be something that is freely entered into and freely left if it proves to burdensome and unrewarding. For example, there will be a strong preference for a volunteer professional army over a perpetual military caste or conscription as you might see in Lawful government. Much more strongly than Lawful Good, there will be the expectation that its really the spirit of the things the law exists to protect that is important, and that rulers ought to be put in positions to exercise their judgment with regard to that unburdened by necessities of tradition or written law - but not so much so that failure to abide by the law becomes and excuse by those in power to abuse their authority to their own advantages. As such, the neutral good government - however it is set up - will be most marked by its checks and balances.
Now of course, in the real world no plan survives battle. All of the above tend to drift in very predictable ways. A classic example is Lawful Evil societies are easily subverted by Chaotic Evil rulers who insist that the society exists to serve their personal whims rather than what is actually good for the society as a whole. Likewise, chaotic neutral societies are easily subverted by powerful individuals who then create laws and institutions that serve their interests at the expense of the personal freedom the society is supposedly organized around. Lawful good societies are easily victimized by incompetent rulers who aren't able to fulfill the needs of their offices. Chaotic Good societies because they see the law as flexible and evolving, can end up generating such a huge body of laws in the name of 'common good' that those laws themselves become every bit as stifling of the personal freedom they supposedly admire as a dictatorship would be (see ancient Athens for obvious historical precedent). And so on and so forth. And as I said, any system larger than a single village in the real world is going to have features adopted from several cultures either by deliberate compromise between several factions or through unguided evolutionary processes. If we really want to discuss real world examples, I'm not interested in discussing anything larger (or more controversial) than the alignment features of the government of individual Swiss cantons or villages.
