Speaking of, can they pick through my discard pile? I know the top card is face up.
I don't see why not. There's nothing to prevent them from writing down what cards you've played, so not allowing them to do so just encourages weird behavior that accomplishes the same thing.
It happened during the last session that one of the heroes asked to look through it before their turn. I allowed it because, like you, I've never heard of a rule against it- it would be different if the cards were discarded face down- that would, to me, imply that it was off limits.