Hello, Mr. P.
I worked in OSJ and it has a special place in my heart. The architecture is trance-inducing and there is excellent live music most weekends. With that said...
1. Yes, I'd say living in OSJ is a good idea in spite of its drawbacks, which I've written a fair amount on in other posts on this site, such as the legion of bums defecating on front doorsteps, the water shortages, the parking problems, the traffic jams, the derelict properties that spawn hordes of mosquitoes, some carrying dengue, etc. I can say that in spite of all of those things, OSJ is still a wonderful, enchanting place to live (if you don't have small kids).
If you go without a car, OSJ is definitely a good place to live because that is where the central bus station is located and there is also an express shuttle to the tren urbano, which has a stop at UPR Rio Piedras.
2. Having a car in OSJ is too much of a pain if you don't pay for parking in one of the three parking garages. Street parking is a nightmare and has gotten much worse since the administration decided to disallow people to park on the road north of the capitol building.
What I think is best is to own a car, both for convenience sake and to be able to properly explore the island, and to pay the monthly rate at one of the parking garages. You should choose which parking garage only after having settled on where in OSJ you will live, as two are side-by-side by the harbor on the bottom of the hill and one (the subterranean one) is on the top of the hill right near El Morro (the NW corner of the islet of San Juan). The distance between these parking garages is too far to not reserve the one closest to where you live, especially considering the heat.
If you choose la Doña Fela parking garage (known colloquially as simply "la Doña Fela" - at the bottom of the hill besides the harbor, near the Bankruptcy Courthouse/Post Office), I'd strongly urge you to pay for a reserved spot because this garage fills up all the time and you will be SOL without your spot. I've spent an hour driving around OSJ hoping to find a spot and the cops are quick with the parking tickets.
I don't recall off-hand the costs of a reserved spot but I believe I paid $90 a month for a reserved spot. The parking garage adjacent to la Doña Fela is newer and much quicker to get in and out of but it closes relatively early (I believe midnight).
3. $6K will certainly be enough unless you need to purchase a car as well. Electricity is not cheap and you may be required to make a hefty deposit in order to get your account activated. Rent is not cheap either (although it could be worse). If you don't have a roommate, look for $1K - 1.3K for a two bedroom in OSJ as a reasonable range. There are a few apartments cheaper but they are usually in bad shape. You will also need to pay the one month's deposit.
4. Clasificados is a very good site but I found that the best way to search for a place in OSJ, be it an office or an apartment, is to walk around and look for advertised vacancies. There are many large signs well-placed on balconies and walls where there are places to let. Then call the number advertised on the sign. This is a surprisingly efficient way to find a good place to stay in an area you like. You could touch base with ten different realtors in one morning doing this.
5. The internet is relatively consistent. There were many outages and the firm where I worked but usually it was fine. One thing to consider that is unique to OSJ: because the walls of the colonial buildings are so thick, intra-office wireless internet (and cell reception when you are in a building) are awful. If you are a heavy user of a wireless router, you may need more than one to cover your apartment.
My last bit of advice is to be very conservative with your spending. PR has a way of screwing you over unexpectedly, for instance, UPR may completely f-up your financial aid. As a general rule, you will be screwed almost any given time you must rely on someone else in PR to get something right for you the first time. This is especially true regarding (1) money and (2) bureaucratic processes. So be wary of running down your cash because you are expecting your financial aid (or anything else) at a certain set time.
Good luck. I think you will have the time of your life. Let me know if you have any other questions.