Seems like rebuilding such a facility would be more beneficial to the city of B'ham than a dome, JMO.
Perhaps give it adequate interior space to serve as a convention hall?
For Phoenix: (the others can fill in the gaps)
Restaurants/Bars:
The Garage - GO! It's one of the coolest places in town, though kind of awkward to get to. Find someone to take you the first time.
Makarios - Funky little building in the 5 Points South area w/ a bright green roof. Great hummus and nice folks.
Jacksons - SOHO area of Homewood. A little too loud, but great food.
Jim N' Nicks - If you like barbecue it's really good. Not really my thing, though.
Rojo - It's in the Highlands. Kind of a neat place but usually overcrowded. Good food, though, usually.
Neighborhoods:
The Highlands - Cool area. convenient to Southside & 5 Points South, lots of parks and architecture that isn't just a bunch of Victorian crap.
Crestwood - some neat architecture, but the edges of the area are dicey.
Southside - most of the bars are in this area and UAB's located there.
Homewood - town just the other side of the mtn. from B'ham proper. Nice area generally... just don't get west of I-65.
Mtn. Brook - mcMansions and bougie folk, but a really nice area.
Don't Go: (as told to me when I moved here)
5 Points WEST
Fairfield
Ensley
Woodlawn
...and avoid most everything north of the tracks by more than a few avenues.
...I'd also avoid 280 altogether if I didn't have to get on it. seriously ridiculous traffic. I have friends who live down it or else I'd never use it.
Attractions:
Vulcan - you can usually have the park almost all to yourself, take friends and picnic. Great place for that. Neat little museum there as well. If you're not scared of heights go up to the top of the statue and look out from the little observation deck. There are some great views up there. I believe there are also some walking trails nearby.
5 Points South - Sort of the main entertainment district in town.
Sloss Furnace - They have events there sometimes, and it's interesting to photograph, but it's an old industrial site, basically.