Hey Josh,
I went to school two hours away in Mobile, Alabama and worked there for a couple months at Tulane in N.O when I first went on active duty with the Army. I would say to take a Streetcar ride down Saint Charles Avenue and back. That's the only route I'd go, don't bother going up Canal Street or along the river.
One good place alot of locals, and some tourists go for breakfast is the Trolley Stop along Saint Charles Avenue. I don't eat Seafood, so outside of that suggestion, I can't recommend restaurants. I enjoy bbq, and it wasn't that good in N.O.
If you like to run, check out Audubon Park. Even if you don't, it's a nice place to walk through, and across from Tulane. That and Magazine street may have some shops to go through.
I actually just enjoyed walking through the French Quarter and looking into the houses and courtyards. When you walk through the streets, the part of the house you see is actually the back of the house. If you want to hear jazz, you could probably hear some on Royal Street closer to canal street.
For Jazz, and where the locals go, you'll want to go to the Fauborg Marigny. There should be some bars there that play jazz. Two things you can find in N.O are beignets, which I'd go to Cafe Du Monde for. I'd also get some Pralines while there. Personally, I liked them from Laura's Candies along Chartres street the best, but they're all over.
You may try a ghost tour there. One place you may look up prior to going is the Lalaurie Mansion on Royal Street. I always stop by and stare when I'm in N.O.
A word on safety, as I'm sure you are aware, N.O isn't the safest place, but don't let that scare you. You'll be safe throughout the French Quarter,, most of the Central Business District, and along Saint Charles Avenue and should be fine a block North of Saint Charles and a few blocks south. You should be fine in a little ways in the Marigny as well. Sections can change like a checkerboard, so one minute you'll see huge houses along Saint Charles and a block or two up and it's run down, but at least you have a good indicator of when the place turns bad. Enjoy it, it's hard to not enjoy the city. Expect it to be somewhat unclean and you won't be dissapointed. It's the one city I don't mind that it's dirty. It has somewhat of a do as you please attitude. You can drink on the streets throughout the city (double check that when you are there), but I have never seen any issues with drinking all over the city on the streets.