You should stay at the Ace Hotel in the warehouse district. It's near a lot of the more popular restaurants for locals, it has its own really great restaurant and bar, and you're a quick Uber or streetcar ride from the Quarter when you want to go. Because you said you'd get tired of Bourbon quickly (which we avoid like the plague here), it'll have you just far enough removed from the out of control tourists. You'll also be very close to the Superdome - walkable in my definition, but people from places where walking isn't as popular tend to think it's too long (Google maps says 0.7 miles). And it's a couple blocks off a restaurant and shopping area called the South Market District - not historic, but convenient when you're traveling.

Answering the question about getting to Tulane for a game, it would take you about 30 minutes on a good day on the streetcar from Canal Street, down St. Charles, and to the front of campus. Going from the Ace Hotel cuts 5 minutes or so off. From there you'd walk 5-10 minutes to tailgating and 10-15 minutes to the stadium. Campus is small, but long and narrow. The plus is it costs you $1.25 for the whole length and you'd get to enjoy scenery on the way. Traveling back you'd probably be ready to relax, so I'd take an Uber or Lyft from the back of campus on Claiborne and it'll take you 10-15 minutes max.

Plenty of food around Tulane, but locals don't frequently go out to eat Creole or Cajun food, so it's probably not what you're looking for. The two centers of restaurants in walking distance from campus though are near the corner of Maple/Adams and Freret/Valence - check them out on Google maps. The former has some great restaurants in Ba Chi Canteen (Vietnamese is big here) and Satsuma, and the latter has Company Burger, High Hat Cafe, Wayfare, and Dat Dog. People's recommendations above are great though.