I think you've allowed yourself plenty of time, so it shouldn't be a plan for failure...unless you've picked Monday, October 9 as your day...that's a government holiday and I think the Archives is closed.

From DCA, you're best bet is to get on the YELLOW line and just get off at the Archives stop. No need to get on the Orange/Blue line.

Since the Archives is close to the eastern-most end of the National Mall, and if this is going to be a walking trip, you could walk the Mall clockwise...heading toward the Capitol Building, then back west toward the Washington Monument, stopping at any museum that interests you. The Washington Monument is ticket-only, and by the time you get to the ticket kiosk, you're probably gonna be out of luck. The White House is due north of the Washington Monument, so may as well head there (I'm purposely not sending you down all the way to Lincoln Memorial...there's another thread in this forum that discusses a similar itinerary and it's mentioned that the Mall is 2 miles from end-to-end...not sure if you really want to walk that much since you mentioned a couple of specific things).

Getting back to DCA, you could make your way back to the Archives stop, or just get on at any Metro stop and make your way back to either the Blue or Yellow lines (transfers are free on the Metro subway).

Where to eat? Assuming you've did the Archives first, you could do lunch at the National Museum of the American Indian. Pretty good food, very clean, and usually plenty of seating. I would avoid the street vendors 'cause they're pretty expensive...for street vendors.

Can't really help you with the airport. I take the train into the city.