In The Three Trials, if I complete my staging, and then use Asfaloth in my Action window before resolution to clear out the location guarding the Key, the quest card instructs me to immediately advance to a new stage 2A. But I haven't yet done quest resolution. Do I complete the setup instructions on the new 2A & 2B, and then do resolution comparing my willpower to the staging area threat, even though that might have changed with the addition of a new Enemy and/or Location?

I think you do it on the new stage, which sucks.

If a card effect forces you to change to a different current quest, like in Battle of Five Armies of many of the scenarios with side quests in the Angmar Awakened cycle, you resolve the quest at the new quest stage. This should hold true if you advance to a new stage using a method such as the one you have described, as well.

The same thing can happen if you use hands upon the bow with Legolas and he places 2 progress as a result of a kill. I've seen it many times.