I'll answer your questions:
1. There's no actual law school gym, but there are a couple options. First, there is the option of paying for the business school gym (which I don't do). Apparently it's nice, but it's also pricey. Next, there are a few free gyms, which you can basically go to depending on location (CCRB, IMSB, NCRB). IMSB is probably the closest to the law school, but I've only been to CCRB, which is the main gym (and honestly, it's not very nice). If you want a nice gym, you might as well fork over the $$$ for the b-school. I'm pretty sure all of the gyms have indoor courts, but I'm not 100% on that.
2. If you want to be close to campus, I'd recommend the replacement Lawyer's Club housing. A lot of the closest, walking-distance stuff is gone already unless you are willing to pay $$$$ (which you may be willing to do). Also, unless you do pay $$$$, the housing is kind of gross. If you have any more questions about housing, PM me and I'll send you some links based on your situation.
3. Things are social and people go out and law students date. It isn't different here than anywhere else. Lawcest is generally gross though, and tends to cause problems.
4. Both groups have a ton of programs, and they often do stuff together. If there is a skew, I haven't noticed it.
5. It depends. I wouldn't want to be here without my car, because it can be difficult to do stuff like grocery shopping, etc. without it. Also, you have the possible option of not living right on top of campus if you have a car. That said, if you don't want to have your car you can probably get on okay without it (especially if you have a roommate that has a car)