I usually hit the bar at Proof or Zaytinya before a show. But get there on the early side - if it's a 7:30 show, seats are hard to find by 5:45. Before the Van Halen show last night, I got shot down at Proof at 5:40, before grabbing the last two bar stools at Zaytinya at 5:45. It was wall-to-wall people from then through 7, and then everyone left at once...
Wherever you go, if the show at Verizon Center is a popular one make sure you make reservations. We stayed at the Hotel Monaco a year or so ago, and had a heck of a time finding a place in the neighborhood to eat dinner one night, as unbeknownst to us ahead of time, Billy Joel was playing at the Verizon Center. All the restaurants for blocks around were booked solid!
We did eventually manage to squeeze in at Bistro D'Oc, where we'd eaten the day before.
Wood burning oven pizza: Ella's or Matchbox (get the sliders at Matchbox, and the spaghetti & meatballs at Ella's, in addition to the pizza at either)
BBQ: Capital Q
Noodles and Dumplings: Chinatown Express
I've also heard Burma is good for Burmese but haven't been yet