I didn't have time to read all the replies, so hopefully I'm not simply rehashing what's already been suggested. I hope this helps.
1. How to stop the PCs from going to the authorities for everything?
There are a number of ways to do this. Here are a few:
A) The authorities can't handle the problem directly, and as a result have actually hired the PC's to handle it. This can be due to political or practical reasons. For example, it's a highly sensitive matter dealing with someone who is very powerful and influential, and thus they're utilizing outside help to avoid political problems falling onto the shoulders of the guard. Another example, it's something FAR more powerful than the authorities can handle and that's why they need adventurers. Some other examples, the city is crime infested, there are riots going on, there is a plague, etc. What it ultimately boils down to is that the authorities can't deal with it, and that's why the PC's have to do it, most often with the authorities permission (though not necessarily always).
B) The authorities are corrupt. Not all of them have to be corrupt, of course, but the point is to make it so the players don't know who to trust. They can even have someone on the inside that they think they can trust (whether they can or not is entirely up to you), but like with A above the insider can't really help because of the corruption.
C) The authorities don't believe the players. Maybe the authorities don't like adventurers who come to town causing trouble. Maybe they're dismissive and don't believe the players when they run to them to inform them about the evil cult in their midst. After all, the PC's just arrived and the authorities live there - it's their job to root that stuff out. Surely they would have discovered it before the PC's. It can also be used in conjunction with B.
D) The PC's are working for someone who wants to keep their business on the down low. Maybe they're handling a sensitive matter for someone important, or maybe they're carrying out illegal work. Either way, going to the authorities isn't really an option.
E) The authorities will want what the players want. If the players are after a powerful artifact that could be useful to the authorities, then the authorities themselves might want it. Keep in mind that even if the authorities don't want it themselves those in power might use them to try and get it.
F) There is no central authority for the city. Perhaps it's run by a merchants council, and the wealthiest merchants of the city each employ mercenaries which double as city guards. This makes it harder to easily go to one or the other as there is no clear analogue to the modern day police officer.
G) The players are criminals and / or they've been framed for a crime they didn't commit. Perhaps they were poking their nose somewhere they weren't supposed to be, and now someone in power is trying to silence them for good. Either way, the authorities won't be of any help - they're out to get them.
H) All of the above and more. The above just scratched the surface.
2. Details, Details, Details -- my brain will fry trying to keep track of names, personalities, locations, and their relative positions to one another, and so on.
Short of extensive note-taking and maps, anyone have any other suggestions that have worked for them?
Rough outlines and learn to wing it. If you know your players are going to be in a city for awhile, then it's important to know some details about the city. That means notes. You can either come up with all the information yourself or use an already designed city. For example, you don't have to play in the Forgotten Realms to use Waterdeep. You can use it as a template and then make radical changes to make it your own, and unrecognizable to players who might be familiar with it.
That's generally how I handle things. I get the basic idea for the city in mind, then I search around for useful material that I could "borrow". Even if I don't use any of the material it will often inspire me with an idea or two. It doesn't even have to be a city for D&D.
You're going to want to start with the size of the city, the type of government, its general location, and culture. For example, a city that is near or in a desert will have special concerns: it's obviously hot so they're likely going to want to wear lose clothes, and they're going to need a way to get access to water.
Just running off this idea here is what I come up with just off the top of my head:
The city is a Magocracy run by sorcerers. Centuries ago a powerful blue dragon had a cult that worshiped him as a god, and he led this cult into the desert to be close to his lair. This dragon impregnated a select number of women among his cult. His half-dragon children displayed a high affinity toward arcane magic - often becoming sorcerers. Nearly 175 years ago the dragon was slain, though this information is not known by the people of the city as it was covered up at the time. They believe their dragon god ascended to the heavens and will someday return to them. The city is divided up into a caste system. At the top are the sorcerers, in particular those who have descended from the dragon. They are privileged, bordering on sacred, and akin to nobility in other lands. They serve not only as sorcerers that are nobles but also the priests for their (false) deity. Their ranking among their fellows is determined by the amount of talent they show in sorcery. The more powerful they are the higher up the ranks they rise. The most elite lead the faith, and among them nine are chosen to form the leadership of the city - which is similar to a council. Non-Sorcerers also have their own castes. Slaves are permitted and they are the lowest caste. The city is located along a desert coastline, and though the sea is extremely salty the sorcerers use a secret arcane ritual to purify water for the city. The ritual is considered divine, handed down to them by their dragon god. There is no central authority, each sorcerer is allowed to employ guards, and these guards are expected to enforce the laws of the city. It is illegal to kill a sorcerer by spilling his blood, and thus the most common form of execution is asphyxiation or poisoning.
These are just some thoughts that form off the top off my head. From there you branch outward and build onto it. You have the leaders of the city. Who are they? There are normally multiple power groups within a city, what are they? In this case, I'll say that the city is primarily a trade city - being on the coast and all - and thus there is a strong merchant caste which has political influence.
Create a number of factions within the city, and give them opposing goals. If an NPC is worthy of being given a name, then that NPC needs goals of their own.
Finally, when you have a rough outline - I'd say two to five pages is enough (though its completely possible to do it with less) - you need to stop building. You need to come up with the adventure hooks for the PC's. Why are they there? If they're already existing characters then how can you establish connections for them within the city?
From there you do the same thing - you start with the seed of an idea and build on it. Enough where you can have roughly one adventure ready to use right out the gate, and the first adventure should be primarily to introduce the PC's to the city and get them invested in it.
3. Mass NPC conversations. In a major urban environment there will eventually be points where lots of NPCs are gathered and discussing things (i.e. The PCs walking into a merchant council and overhear them talking about their latest problem that is blocking trade, etc). But I feel awkward when I'm acting out 2 or more NPCs talking to each other because it turns the players into an audience rather than participants.
When handling large NPC groups I find it is best to use a form of shorthand. The PC's don't need to know every word that is said, only the important details. An example:
"As you enter the council chambers it is packed tight. Every seat is taken and you need to stand near the back. There is a great deal of tension in the room, most of the faces you can see vary from grim to angry, all eyes seem to be focused on one man at the front. He is a stout and balding, though still clearly in his prime. Dressed in the plain clothes of a commoner, he is shouting angrily and loudly.
Bald Man: "...your tariffs will be the death of us! The ships won't dock! They sit out in the harbor in protest!"
Councilman: "We understand your concerns, Gregor, but we must collect taxes. The city has been in debt ever since -"
Crowd: "There is hissing and booing from multiple people in the crowd. Someone shouts a profane insult. Another man stands up and spits in the general direction of the council before turning to leave - only to be arrested by the guard. The tension in the room rises, becoming nearly riotous..."
So forth and so on... At any point in the exchange the PC's have the opportunity to jump in and give their opinion. Perhaps they support the Council, or perhaps they don't - either way they've learned the city is in debt, and that provides useful information.
4. I am not a good spontaneous speaker for NPCs. With so many NPCs and so many possibilities of who and what the PCs can discuss, I'll have a hard time coming up with responses on the fly to PC questions/conversations. And, the people I tend to game with are generally better improvisational (sp?) speakers than I am.
This can be a bit difficult, but this is also where it helps to have a good understanding of your city. Having a notebook or campaign journal is very useful as well. Try to determine what your PC's are interested in doing and learning about. You can then focus your building efforts in that direction and be more prepared before hand.
5. Spatial considerations - combat areas simply need to be larger than is convenient. In a dense urban area, I'd imagine the interiors are smaller, leading to either really cramped combat areas, or insides that are bigger than the outside, or every building just has some basement/underground type area as a default standard.
I guess I could just make underground areas a default standard building feature... or if the city is specifically defined as being magical perhaps interior space is magically larger. Or are either of those ideas just too weird?
It depends on the city. I don't use miniatures so I can't really help you there. However, I will point something out: witnesses. It's not like a dungeon crawl there are people pretty much everywhere. I also want to point out some of the useful improvisational items that the players can use. For example, let's say a tavern brawl breaks out. Well the players could flip over a table and take cover behind it. Or they could use a chair as an improvisational shield. Or perhaps they use an ale mug as an improvisational weapon. Things like that. Players may tend to use such tactics since the area is easier to visualize in their minds because it is more familiar to them. After all, who really knows what they could reasonably have access to in a dungeon?
6. Prisoners - when it's monsters it's easier to say kill the enemy in combat. when it's humanoids, it is more likely to result in the players (via the PCs) having drawn out discussions in every combat as to whether to kill or take prisoners. And while I can appreciate the role-play aspect of it, if the party doesn't have a general consensus it will just turn into a conversation that gets repeated way too often for my liking...
Is that something I should just try and setup/get consensus on in the campaign start-up or really just let it evolve on it's own?
This is really a group dynamic issue. It also depends on the situation. There are also legal concerns, after all, the PC's can't just go around killing random people generally speaking. All of that needs to be factored in, and made clear to the characters - that makes such decisions easier.
However, if the players decide to do something like take prisoners, don't spend a huge amount of time on that unless its necessary. Just hit the fast forward button and have the players hand them over to whomever, and be rewarded properly - giving the players a summary of what happened. I'd only play out the scene if the group REALLY wants to do it. Generally, though, when someone wants to play it out they just want to ask a certain question or something - you can just tell them what they were told. Don't let yourself get bogged down.
All that said, maybe I should just avoid having such a strong city-based campaign since I know it is a weak point for me. And instead stick with what I can do better (and leave the cities for side-treks rather than the main focal setting) ?
No, I think you should do one. It doesn't have to be a long campaign, but it doesn't hurt to try it. After all, you can't improve if you don't practice.