by shnar
1) Each player gets to make one voluntary engagement. Then the first player brings down every monster that has a level less than the first player's threat. Then the second player picks up any stragglers, following threat limits, etc.There are two big advantages: As Terefor pointed out, engaging an enemy removes it from the Staging Area and so its threat no longer counts when questing. That can be useful. Secondly, sometimes the first player simply doesn't have the fighting heroes/allies to handle all the enemies that might come down from the staging area, so having player 2/3/4 voluntarily engage an enemy helps spread the damage around. For example, in questing, you reveal 4 enemies with an engagement level less than 20. It's early in the game, and 4 enemies would crush player one, killing his one ally and 2 of his heroes. So players 2 and 3 voluntarily take 2 badguys to their area, now Player 1 isn't so much under the gun and survives to fight another day.
2) If Unexpected Courage allows him to refresh from being exhausted, yes (I don't remember the text of UC).
3) The Hero isn't really engaged with a badguy, the player is. When attacking, you exhaust all the characters that you are attacking with, exhaust characters and make your attack, then look at the villains left in your engagement area and repeat. If you have a card that lets you ready an exhausted hero, then that hero can attack again on a different badguy.
-shnar