I was a resident of New Orleans for 10 years. I left in 2011 and haven't visited since then, so my knowledge isn't the most current.
BUT a lot of these suggestions seem really expensive/overly touristy to me.
If you drink in the French Quarter you will be paying way too much. Try Frenchman Street in the Marigny for a more relaxed/cheaper experience. A lot of the bars in the FQ have drink minimums - ie, you must buy two drinks per set of music. If you DO end up at Pat O'Brien's, note that there are three different bars - the outdoor bar with the fire fountain, the piano bar, and the nondescript bar with the big TV. Drink prices are DIFFERENT at each bar and cheapest in the TV room (it's to your left when you use the St. Peter St. entrance). You can take your go-cup from the TV bar into the Piano bar. Hurricanes are unique - don't spend the $ on one unless you have tried someone else's first and know you like it. And ask for a plastic cup unless you really want the glass souvenir glass. You pay for that glass anyway. If you do want it, the dude by the door will wrap it up and box it for you on the way out. If you don't want it, you just left $2 or $3 on the table. That said, Pat O's has one of the nicest bathrooms in the FQ and it is worth the $1-$2 tip to the attendant.
Preservation Hall is a good suggestion for traditional jazz. That and One Eyed Jacks are about the only places I would go in the FQ for entertainment.
I like the Gumbo Shop for inexpensive gumbo. They have a yummy vegetarian gumbo too. I second Gimesalot's other suggestions for inexpensive food in the FQ. And you kinda do have to try Cafe du Monde. Don't wear black when you go - there will be powdered sugar EVERYWHERE.
I'd head uptown (you can take the St. Charles streetcar) for cheaper eating in general. A streetcar ride up St. Charles is fun - tons of interesting architecture, Audubon Park, the Latter Library branch are all free for the eyeballs. Magazine Street has a bunch of restaurants and fun shopping - it runs parallel to St. Charles, between St. Charles and the river. Tipitina's or the Maple Leaf Bar are the two music venues I'd recommend uptown. Jacques-Imo's is a super-fun restaurant near Maple Leaf, but they don't take reservations unless you have at least 5 people, and you can wait up to two hours if you don't have a reservation.
It will be crawfish season and you should try boiled crawfish if you never have before. I'd go uptown to Big Fisherman seafood market and buy a pound, then take it across the street to the Bulldog bar and get a beer to have with the crawfish on the Bulldog's outdoor patio. Get a local to show you how to peel and eat them, then go back for more if you like them. (If you buy oysters or crawfish at Acme Oyster House in the FQ, you will pay 3x more.) Or go get a poboy at Domilise's or Parasol's. Or get a sno-ball at Hansen's - make sure you try the condensed milk if you like creamy cold treats.
If you do want a splurge meal, and you are bringing *dressy* clothing, try Commander's Palace or K Paul's. Don't try Antoine's - it is interesting because they are the oldest continuous operating restaurant in the U.S. and they are still serving their original menu (and it is like a museum inside), but the chances of spending a ton of money on a dissatisfying antique French meal are very high.
I wouldn't go to Avery Island. There are so many things to do in the city proper that you might kick yourself for spending the money to get a car or pay for an expensive tour bus to get there.
You might like Backstreet Cultural Museum if you are staying in the Treme. Lots of Mardi Gras Indian memorabilia and it's not expensive.
Music - the only two types of music that are SOOOO "New Orleans" that you can't miss them are Traditional Jazz and Brass Band music. Trad - look for the Loose Marbles, Palmetto Bug Stompers, Jazz Vipers, Preservation Hall. Brass Band - look for the Hot 8, Soul Rebels, Rebirth, Free Agents. I can also give a shout out for high quality entertainment courtesy of a couple of my friends:
Burlesque: Trixie Minx (a former professional ballerina with an amazing stage presence - she performs all over the world):
http://www.fleurdetease.com/shows.htmlStylish Jazz: Jayna Morgan has a few bands. Swing Setters play kids music, Sazerac Sunrise Band plays swing and trad jazz, and she just formed a new band called the Bella Donnas which is composed entirely of women. I think they play trad jazz too but am not sure - their debut show is this weekend.
http://nolaswing.com/https://www.facebook.com/TheBellaDonnasNOLA/Transportation from the airport: try the bus, or maybe Uber. Regular taxis cost $40+ one way to get into town (the airport is in a far-flung suburb). Taxis are super-cheap in the city proper, and there are pedicabs too. Don't even think twice about calling or hailing a cab if you are drunk and need to go a mile to get to your hotel - just do it.
Some final notes on safety: New Orleans is not a safe city. You will need to bring all your street smarts, even for touristy areas like the FQ. The Treme isn't the safest part of town. DO NOT wear Mardi Gras beads. If you want to keep some, put them in a pocket or a purse. They will mark you as a tourist and you will be a target. At least half of my friends have been mugged walking home to their places in the Marigny, Bywater, Treme areas. Some have been knocked off bikes to be mugged, and had their bikes stolen too. The police will take an hour or more to come if you call them - they are stretched beyond endurance right now. I even know someone who was shot in the face right on St. Charles Avenue, which is generally considered to be safe. If you have second thoughts about anything, choose the safer option. Say "New OAR-lins" instead of "New or-LEENS" if you want to sound more local. PAY for a GROUP cemetary tour if you go to any of the cemetaries near the FQ/Treme. Tourists are definitely targeted if they try to explore alone.
But have fun! It is a wonderful place to visit :)