I did my undergrad at Vandy, and I'm 99% likely to be back this fall.
Vandy's made a big push in the last few years to have all the undergrads living on campus, so that means that A) I can't really answer your first question, and the undergrads are fairly well contained.
From what I've been researching and know from past experience, living right next to campus gets pretty pricey, but it drops off pretty quickly as you go further out. Vandy's in a nicer part of town, and there are plenty of places to live nearby. I'm planning on finding something bike- or walkable. There are plenty of places to eat nearby, but grocery options are limited unless you can drive out to the bigger stores. There's a grocery store on West End near campus, but it's nothing like the stores a suburban kid like me is used to. Of course, I haven't lived there in 4 years, so maybe it's changed?
As for the rowdiness...the undergrads, like I said, are mostly contained on campus. Certain areas like frat row can get pretty bad on the weekends, but it's all on campus and not near any off campus residences. With the addition of the freshman commons after I left, I'm sure the feel of the different areas has changed now that there are more upperclassmen in formerly freshman dorms, but I can't imagine it's changed all that much. Otherwise, I have to say (having been one) that the undergrads are largely a sharp group.
Lack of diversity's an issue they've been trying to tackle, and you get plenty of people from southern, monied backgrounds, but I think it's something that's been improving. (My group of friends looked like the Small World ride at Epcot, so it wasn't an issue for me.) Owing to the wealthier, southern background, the student body tends to be a little more conservative than you might expect at a private university Vandy's size, but it's not in your face (like it is at the university I work at now).
Hope that answers some of your questions. I'm sure a native Nashvillian or one who hasn't only lived in the Vandy bubble has a different take on things.