So I wouldn't be at any disadvantage by attending Vanderbilt rather than NYU if I wanted to match at NY Presbyterian or other programs in the Northeast? Any other comments would be greatly appreciated!
Yes and no. If you're a good student at a place like Vanderbilt, you will absolutely match at the big universities with no problem if that's what you really want to do. Doing an away rotation is certainly helpful, particularly for California programs, but by no means necessary. It also depends on why you want to move to the Northeast- is your family/SO there or do you just kinda want to live in NYC? If it's the former, and it's clear in your application somewhere (you did a lot of schooling in the NYC area, your permanent address/ birthplace is around there) you shouldn't have a problem. A lot of interviewers ask about ties to the area when you're there, so it's nice to be able to say "my family's here, I'd really like to come back" instead of "nope, but the area seems cool".
That being said, it's better the devil you know than the devil you don't, so the big name East Coast places might have a little bit of an edge, particularly if the program is very small. Agai, this has a lot to do with ties to the area. If it's the in-home program of a med school, they have the added benefit of having worked with those students/ the people who wrote their letters of rec, which really helps.
But a place like Vanderbilt? No, that won't close any doors.
Do medical students have an edge applying to the residency program at the hospital associated with the medical school they're attending (if there is one)?
Again, yes and no. That depends a lot on how many people from your class wants to get into the home program for that particular specialty, because programs tend to cap the number of residents they take from their own med school. On the one hand, most programs I've heard of will interview their own med students and anyone who rotates through there regardless of board scores/ how competitive you are/ whether you actually want to go there. At best, it's a way for you to rank (and hopefully be ranked by) a program that you may otherwise not have gotten a shot at because of below-average scores or something. At the very least, it's a practice interview so you can become more polished later. So to that effect- sure, it's a benefit to come from that school- you'll get an interview no matter what, you have a chance to get to know the faculty and program director, the residents know you well, etc. But you'll almost always have to deal with a cap on the students they take from their own med school, if you pissed off the residents during your time there it'll be problematic (maybe you wouldn't have put your foot in your mouth in a 6 hour interview day, but you did during the months you were rotating on that service), and you'll basically be compared head-to-head with your classmates who are applying to the home program.
What would make doing an away rotation not a good idea? I thought it was a good way to see if a particular program was a good match for you?
See above. The problem is that they're also seeing if you're a good match for them. On an away rotation, you're expected to work harder and longer hours than that program's own med students, cause you're there to impress. It's a one month long interview for the job. You can't just say "my only weaknesses are that i work too hard and care too much" during your interview and get away with it, because they've seen your weaknesses first hand. It's also generally much easier to piss someone off than impress. I dont know about you, but in one month I can put my foot in my mouth multiple times, but I can generally keep it together for an interview day.
That being said, as WS pointed out, there are some specialties where it's basically required to do a bunch of aways, and sometimes you pick a specialty your med school's hospital doesnt have and you have to do aways regardless. Most people do fine. You just want to be the best they've seen while never making anyone angry or making them think you're not on par with their other students, and that's hard work.