is there a shuttle bus

Yes there is, a burgundy shuttle bus located at Foggy Bottom Station

Yes! There is a shuttle bus that runs to and from the foggy bottom metro station before and after shows.

Yes

They do have one thats every 15 min m-f from 9:45am- midnight. And then Saturday starts at 10am- midnight, Sunday is noon-midnight and federal holidays 4pm-midnight. They are on left side of eacolator exit on New Hampshire Ave from the 23 & I St NW metro stop.(froggy bottom/GW, orange , blue, and silver line) Which is closest subway/metro stop then a short walk to Kennedy center.

Yes. It runs from the George Washington hospital subway station.

Sorry there isn't public transportation........