River mucking situation in home game

[AJo Go All In]

Something unprecedented happened in my home game tonight. What should have been the result of this?

Big pot developing, lot of action preflop and on the flop and turn (this is a .5/1 no limit hold'em game). The board is all rags and on the river an ace hits. Player A bets, Player B thinks, and thinks, and calls. Player A says "Nice call" dejectedly and starts making the motion to muck his cards. He then flips his cards face up and throws them into the muck. He had an ace and a jack (this wasn't a slowroll, he thought he had lost the hand). Player B then starts to rake in the pot. Now A asks what B had, and B shows pocket kings.

What if A's cards had not yet touched the muck, and B, as a courtesy, had shown his hand?

What if A flashed his hand and then threw it in the muck?

Does it matter if A had thrown his hand face down in the muck instead of face up?

Some background, this is a home game that is taken fairly seriously, we all know each other, but nobody is softplaying. We don't follow casino rules to the letter per se, but we do (generally) enforce things like not coming in on the small blind, throwing out one chip means a call, string betting, etc.