Disclaimer: answers refer only to my knowledge of events, which is anything hardly complete. I've missed a lot of happenings in more recent adventures.
Is there a powerful criminal syndicate in town? Is it like the mafia? Does it provide social services or just control crime?
Mmm, not really. Early on in L4W, we avoided creating a single powerful criminal organization, preferring instead to let individual DMs come up with gangs on the fly as needed. So there have been a number of separate criminal gangs in Daunton, of varying "power levels", at various times. One assumes that the gangs that have appeared in adventures are just the tip of the iceberg; there are many more where they came from. I have the impression that in Daunton, crime is generally pretty well controlled by the city guard, and when it gets out of control they call for the adventurers. So it's pretty safe most of the time, with the possible exception of the poorer areas.
In Bacarte, the criminal element is a bit closer to the surface, and the consortium of merchants that "rules" there could certainly be described as a criminal syndicate -- except for the fact that they're the ones who make what few laws there are.
Outside of the Near Isles, it's pretty much DM's prerogative. If you want to run an adventure with a powerful, controlling crime syndicate, we can probably find (or create) you a place to do it.
Are any of the particular heads of powerful temples named? If so, are they in the wiki?
Belingras Erigard is the leader of Lauto's temple in Daunton. I'm not aware of any of the other temple leadership having been named, but I may well have missed it.
How does the town guard work? Are the police one body, or are there varying levels of authority, similar to the town police versus the state police versus the FBI? Who controls the town guard? Are the members of the town guard also members of the army, or is this not necessariuly the case?
I'm not aware of any established divisions in Daunton's guards; they seem to be monolithic as far as I can recall. Ultimately they answer (I think) to the mayor, Gunter Brunt, and through him to the rest of the government. To my knowledge, I don't think Daunton has a standing army to speak of, outside of the guards. Daunton is more about defense and survival, particularly against monsters, than conquest. The Imperium probably has the closest thing to a professional army in the Near Isles. Outside the Near Isles of course, there are plenty of kingdoms and empires with armies: Jade and Hzaka come to mind.
The Guards tend to get used by DMs as stock / throwaway NPCs, so I don't think we've defined them in much detail.
Is there a "stock exchange" for lack of a better word where people can invest money and powerful monied interests can interact?
I don't think there's a specific place established as a center of that kind of activity (to my knowledge, of course). Finance in the Near Isles probably isn't quite as advanced as it is here; I doubt there are such things as publicly traded stock corporations. I imagine that partnerships or privately held (family) corporations are about as far as it goes, in that respect. But, hey, what do I know. I think most wealth tends to be concentrated in noble families, and they invest most of it in things like trade expeditions, merchant ships, and such.
If you need a physical location where a lot of financial transactions are taking place, I might suggest Lauto's temple. His portfolio includes trade and commerce, and his temples are known to give loans. I imagine they'd be the place to go for contract notarization, arbitration and such. All for a small fee, of course.