How would these differences manifest in good or evil chaotic deities?
Well, good believes that whatever meaning can be found in life is to encourage health, success, happiness, growth, prosperity and so forth in yourself and in others - what Gygax put succinctly as "weal". Evil on the contrary believes that value is only found in destruction, negation, and tearing things down. In the terms of Chaotic Good versus Chaotic Evil it's literally the differences between a being wanting to enter into an intimate friendship with you, and the difference in a being that doesn't see you as another "self" at all and only sees you as food or other property to be used to its ends.
The primary motivation of most Chaotic Evil deities can be described as 'hunger' in one form or the other. They
want and that's it, and the cult may acquire some power by providing for those wants. I personally feel that NE and CE are fairly hard things to worship in truth, and that most of those cults would be relatively rare and most of their worshipers transitory in a sense - they attract the desperate and depraved. Sincerity doesn't really exist in CE philosophy, so the idea of sincere worshipers is fairly alien to it. Probably some of the most widespread (and frightening) CE cults would be the parasitic ones where the god has basically subsumed the identity of all of his/her/its followers. Cults were everyone is a lycanthrope and the real god is the curse/disease/manifestation for example. Nurgle from the Warhammer world is a fairly well done example of this. On the extreme end of the spectrum, some Chaotic Evil deities might not have any relationship with their cult at all. They might arguably not even realize it exists. They might be incapable of realizing - like the blind god Azathoth of H.P. Lovecraft - that anything other than themselves actually exists.
Chaotic good cults on the other hand are built around friendships and other close relationships. There will be a lot of emphasis on disciple/mentor relationships. The relationship of the deity to the cult will be precisely this, and its highly likely the deity will walk among his/her followers (usually but not always in disguise). Clerics will be trained individually by no formal process, and there may be few or no actual physical intuitions associated with the cult unless the deities particular philosophy encourages the need for a facility. For example, the popular chaotic good deity Jord from my homebrew has his followers build and operate gymnasiums, and so all Jord temples are also gymnasiums and gymnasiums are a sacred space to him. Other chaotic good deities tend to prefer to only worship in sacred spaces in the natural world, or they may be encouraged to see everywhere as a sacred space and discouraged from building temples for that reason.
In a polytheistic society, there is certainly no reason why you can't worship broadly, but I suspect chaotic neutral deities will attract some of the most widespread and fervent worship.
Also how do you feel about the standard portrayal of drow society?
I feel it's complete bollocks, and Salvatore and others have just made it worse. The typical portrayal of drow society is that it is a quasi-fascist, strictly hierarchal, meritocracy with rigorous exams and strict control over drow society collectively. Yes, some nod is made to the idea that they are fighting with each other and contesting with each other, but a close reading of say the culture supposedly pervading the Nine Hells reveals no distinction between it and the culture supposedly pervading the Abyss and Drow culture in particular.
I generally feel that the writers of D&D, especially those post Gygax (although Gygax is not blameless), have had a very poor understanding of what is meant by good and evil and great difficulty explaining it. They can to some extent correctly label the works of evil as evil - they know that theft, murder, and so forth are evil. But the closest that they can come to why those things are evil is that they are 'selfish'. The result is that the three evil alignments are basically indistinguishable. The treatment of LG and CG have scarcely been better, depending on what the particular biases the writer has had with respect to what good looks like.
For my game, I model Drow society in my head very much after the social order of lions - which is one of the ugliest social structures you will find amongst any social society of any animal. As chaotics, gone are these large, highly organized cities with a shared top-down culture that typify drow culture in its normal conception. Chaotic evil cultures are violent anarchies - they can't support that sort of organized structure. Instead, in drow society land belongs to matriarchies that consist of close relatives. These females band together for mutual protection and build small keeps or fortresses to dominate valuable resources in the underdark. Males children are discarded prior to puberty, and females brought up as part of the band. Males must find and win acceptance in the nomadic male gangs that roam the margins of female territory. Many are in fact killed outright. Male bands are in a particularly precarious position. Food resources are scarce so small bands do better than larger bands. However, small bands can be easily killed by larger male bands. Recruits therefore need to be able to pull their own weight and at the same time not threaten the existing hierarchy - many bands dissolve in violent murderous conflicts. The goal of all male gangs is to grow strong enough to successfully assault and conquer a female fortress. The survivors of this can claim breeding rights from the uncooperative females, and kill any young males and non-breeding age females that they find. All breeding occurs through acts of violence. Male bands attempt to defend these breeding "rights" as long as possible - after all, the females are now pregnant with their genetic heritage. The real deepest horror of all of this is the same as it is in lion culture. Females will fight viciously to defend the offspring of prior male bands, because it is also their offspring, but as soon as those children are dead they are basically forced to consent to the perpetual violence including what is inflicted on them. I don't go any further in describing this process because its pretty darn revolting and degrading. Feel free to research lions further.
Anyway, the players ideas about what to expect from drow constitute the myths of my world. And further, while the drow if they ever emerged would in fact be very dangerous, it wouldn't be because they are organized. A plague doesn't require organization to be dangerous.
