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Your second question's answer should be the Jack of Spades. Now, the chess
problem is a horse of a different color. With the setup, black is in check,
therefore, something must have blocked the check. That something should
seemingly be the white king, which, in the b3 position, would have blocked
the white bishop's check of the black king. In that position, white, king
in double check, would have only 2 moves to relieve check, one to c3 and
the other to a3. Since the king fell off the board, and no other piece did,
it is logical to assume the king could not be any farther from the edge
than pieces that did not come off the board. Thus the king moved
b3->a3->floor. 

The question then becomes: how did white get into this position? Before
the king moved, it was in check from both the bishop and the rook. How did
that happen? 

If the white king was on b1, and the black bishop moved from b3->d5,
discovering check, that bishop and the black king force the white king to
the corner, a1, a more logical square from which to fall alone. However,
this would require a blocking piece on c2, a piece misplaced in the initial
problem, or some non-FIDE rule(s), no? 

So, I'll bite. What piece of information, in front of me or not, am I
missing? 

Been a busy spring and summer; haven't been able to do this much at all.
Heck, I've only designed one game all year, and it's just now about to be
playtested to see if it's anything at all, or a colossal flop. Recently
I've been a poor excuse for a prolificist, George. I'll try to do better
in the future. ;-) Enjoy