Part of me is sad when I see that list, as I realize you are only going to be seeing about 1/7th of the country.
There is a lot more to the USA than just the BosNYPhilaWash corridor. Too bad you guys will probably be heading home after the 22nd, as a few side flights to Atlanta, Nashville, Chicago, Minneapolis, Saint Louis, Dallas, Phoenix and Los Angeles would give you a more accurate "flavor" of the country. (And, if you were in Minneapolis, I'd actually have half a shot at seeing your choir sing.)
I guess it'd be just as bad as me only going to Sydney or Brisbane and assuming that all of Australia is like that.
Anyways: it does sound like you're going to have a pretty packed schedule, so I'd suggest doing some of the touristy things: go on a bus tour of New York City and the area when you're there July 13-17; make sure you see the WTC memorial and visit the Statue of Liberty (though it will be closed for repairs to the base). Times Square is now a pedestrian mall, so check that out, along with the Empire State Building.
Not sure if you're a footy fan, but you might see if the New York Red Bulls of MLS are playing that weekend (or maybe DC United, when you're in the capital).
If you guys are in Philly the 18th, try to see some of the major historical sites before your concert; at the very least try to get to Independence Hall, where the Constitution was signed. And make sure you try a Philly Cheesesteak or two.
DC has a LOT of things to do; I'd like to echo Baindhe_dub that you shouldn't try to see all of the Smithsonian museums. Pick some of the ones you'd really like to see - maybe the Air & Space Museum, or the American History Museum, or the main building (the "Smithsonian Castle"). Try to get in to see the Capitol Building and the White House, and visit at least one of the memorials/monuments (Lincoln, Jefferson, Washington).
Your group might be already slated for some group tours; they're easier for those from out of the country to use so that you can hit the "major" sites in your limited time here.
Just keep in mind what I said at the start: you're only seeing a sliver of the country.