If I try to use Rain of Acorns (or perhaps another card that causes my opponent to exhaust a character, can my opponent exhaust that same character for another one of his effects, say criminal mastermind? And if so, do I then get to pick a different character to exhaust?

thanks for any help

They can exhaust the character before Rain of Acorns' effect resolves.

You target upon delcaration, so you cannot choose a new target if they do as you say.

He can absolutely exhaust that character while your effect is on the chain; your effect only exhausts his character at resolution, he's just like any other ready character.

And targets are always chosen as part of playing the effect, directly before costs are paid. You don't get to change your target later.

Note, though, that we're just talking about targets, here, which means effects that have the word target in the text. If an effect doesn't target, you don't make your choice until resolution---Null Time Zone; Justice, Like Lightning, etc.