Okay makes sense to me now. But here is a slightly different situation. How would this work?
My opponent has Amazing Spidey, and I have...let's say Marvel Defenders. It is opponent's initiative. He discards a card to exhaust Beast, who is in the back row. Can I exhaust Beast to Book of Vishanti in response to that, or is he just exhausted/can't do anything about it?
Thanks in advance!

You can exhaust him in response. Before Amazing's effect resolves, you will have already exhausted Beast. Amazing's effect isn't negated, but instead of it trying to exhaust a ready character, it exhausts an exhausted character.

Now, if they had targeted Beast with Professor X 7 from MOR, you would still take the burn, even if he wasn't ready any more. Any effect does everything it can to resolve, so it would attempt to exhaust Beast, and then burn you for the 3 or 2 atk he has or what have you.

Correct to further clarify.

Your opponent pays for his effect - Discard, and then passes to you, you can react by placing your own effects on the chain - exhausting beast, and then you pass back assuming neither of you has any more effects to place on the chain, effects then begin to resolve - Beast resolves - both players pass - Spidey resolves - exhausting an already exhausted character.

So why hasn't Kariggi volunteered to judge for any of my events?

Ok, so this is a related question:

Attacking: When you declare a character as an attacker, does he immediately exhaust, or does that go on the chain? Basically, if someone declares a character is attacking, can you immediately exhaust that character (ASM, Mystical Paralysis, etc) to prevent it?

As far as I can remember, the attack doesnt 'go on the chain' per say, but essentially what you described is correct. When a attack is proposed, the attacker must be a legal attacker (has flight to attack the back row, has range to attack from your back row, ready, etc). When you play Mystical Paralysis to exhaust the attacker, it is no longer ready and thus is not a legal attacker upon resolution of the proposed attack.