So if something says "draw a card, then discard a card..." and I draw a lightning bolt, can I quickly cast it before I have to discard it? I would guess no because it's the entire ability that resolves when it finally resolves...but I guess it doesn't hurt the check.
Correct. Both actions are part of the same spell/ability resolving, and no player has priority during that. You can only take the actions, that the resolving object instructs or allows you to take.
Former Rules Advisor
"Everything's better with pirates." - Lodge
(The Gamers: Dorkness Rising)
"Any sufficiently analyzed magic is indistinguishable from science."
(Girl Genius - Fairy Tale Theater Break - Cinderella, end of volume 8)