Does white still have kingside castling available?
If you mean 1.Rf1+, then 1... Bf3.
May be King at c1?
Yeah f4 is not an option.
Yeah, c1 would be a possible solution, but I cannot see how the position could be reached. f3 seems the only viable solution assuming a castle is possible. without the ability to castle, black can defend ... for a few moves anyway.
Frankly, I don't believe the black king "fell" off the board, I think black slapped it off in a dramatic resignation move...
There are two legal answers.