I attended Tulane and did my MPH there. As far as living arrangements... I don't recommend Demming. It's overpriced for what you get. After 5pm most things close nearby, and the neighborhood is a little sketch to start.
The quarter seems cool, but again, expensive, parking is a nightmare, and you may get really sick of the tourists really quickly. Especially come Mardi Gras time.
Most students live uptown. I personally live WAY uptown, and it takes me 15-20 min to get to the medical district during heavy traffic, 10 minutes during off hours.
Other great areas to think about are Mid-City. It did flood during Katrina, but the area is coming back pretty nicely, and you can get some fantastic deals on nice appts out there.
Also, if you are more into the funky living, consider the Marigny, which is next to the quarter (opposite side of uptown). There are lots of cool bars and great jazz clubs that have not gotten quite as tourist-ified as the quarter.
Wherever you move, I strongly recommend running the area by someone who lives here 1st. More so than most places, neighborhoods can go from great to really bad over the course of a few blocks. You may regret renting a place in an area that seems ok, only to find you are actually living next to a crack dealer (it's happened to me).