Don't know about 'a cracker' used to mean a lie. If an extraordinarily strange or funny anecdote is related, one might say "That was a cracker!"
The cat being called crackers definitely means it's barmy.
I've heard 'crack' used by the Irish to mean a good time, or a good laugh, I think. The only other word I know which closely fits this meaning is an Austrian term 'Gaudi'. Or could one say 'merriment' perhaps?
'Crack' also means a blow, or wallop, which, I guess is why one says 'Give it a crack mate'.