When an effect reads "Exhaust a character", you can pick a character that is already exhausted (example: Shocker's effect). But if an effect says "Stun a character", can you pick a character that is already stunned? (example: Post coming into play)

So though you are able to exhaust an exhausted character in order to resolve an effect, you cannot effectively stun a stunned character.

As long as it is part of the effect, yes. In the case of Post, you could choose an already stunned character.

HeroComplex, it is not "can you can do {foo}" it is, "is something who is already {foo} a legal choice to try to {foo}"

To say you can exhaust an exhausted character is a misnomer. You can TRY to exhaust an exhausted character, but you don't actually exhaust them.

In the same vein, you can choose to TRY to stun an already stun character as part of an effect. You will not actually stun the character, but there is nothing that stops you from trying.

I'm so used to talking about this in terms of costs, I just read too much into the quote. Should have been thinking more clearly on its context. Thanks for the swift kick. :)

If an stun effect is a cost (as in turning face up happiness home), and then I replace the stun with something (example, Toad going to my hand), would the cost be considered paid?

203.8 Actions taken to pay costs can't be replaced. If a replacement modifier attempts to replace part or all of a cost, none of that cost is replaced.

If an effect stuns an already stunned character, would the controller of the character get the damage from the stun?

No. You only take damage as something is stunned. You aren't "stunning" the stunned character, you are "attempting to stun" the character. It doesnt get stunned again.