Hmm. I disagree with some of the stuff here.
Smithsonian museums are the obvious must-do stuff. Taken together, they might compare to the British Museum, but I would place the British Museum as the greatest and most impressive museum in the world. American History museum currently has about half its exhibits under renovation. The industrial age/war/american president exhibits are all open though. But not the pop culture one. Air and Space is the one I'm most excited by. If you're into art, the National Gallery is okay, but not as good as the European art museums, or even the Met. I've heard great things about the African-American museum, though there may be long lines for it since its new.
If you're going to a non-Smithsonian museum, I would suggest the Newseum or Holocaust museum over the International Spy Museum.
As for food around the mall, food trucks are honestly the way to go. There are usually dozens of them lined up. Most of the buildings immediately surrounding there are government office buildings, so actual non-chain restaurants (or cheap restaurants, for that matter) are in low supply within a reasonable walking distance. The food trucks tend to be near the Washington monument, on the White House/Capital building sides. If its a weekday, there's usually another group of food trucks within a block or two of air and space museum.
The other food recommendations are fine, although I'd say Dupont is in general overpriced (unless you go north on Connecticut from it) and you're better off Ubering into AdMo rather than taking the metro to Woodley Park and walking. Old Ebbots is a bit expensive, but there's a lot of history there.
Monuments by night can be fun. Less heat, less crowds, and they're all lit up. If you do them during the day, most can be hit running a more or less straight line going Vietnam-Lincoln-Korean-MLK-FDR-Jefferson. The other two big ones are WW2 and the Washington Monument. If you have time and are interested, you can go across the river to see Iwo Jima, Arlington National Cemetery, and the Pentagon (there's a 9/11 memorial there. I'm not 100% on tours of the Pentagon itself, though I've been on one).
As for other free stuff, the National Zoo is in Woodley Park (Yay Pandas!), the National Archives hold the Declaration of Independence and there are tours of the Capital and Library of Congress that are available to the public as well. Do go see those buildings and the Supreme Court (closest metro is Union Station - a beautiful building on its own). The National Building Museum is also interesting if you're into architecture.
Its not free, but Nationals games are super cheap if you like baseball. And the whole waterfront down there by the naval yard is getting pretty built up. If you like gambling, the MGM National Harbor just opened a few months ago, though that is not metro-able. You'd need to drive/cab. For shopping, Georgetown is pretty nice to walk around and do window shopping, though it can get nuts on weekends, and is a very popular brunch area.
Hope that helps some.
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NHL sucks
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