If a Scheme Twist causes you to KO all of the Super Heroes in the HQ (like the Super Hero Civil War Scheme), do you immediately fill the HQ back up or do you have to wait until your turn is over?
You immediately refill the HQ. From the base set rulebook, top of page 13 :
Whenever there is an empty space in the HQ for any reason, you refill that space immediately with the top card of the Hero Deck, faceup.
In the absence of conflicting rules (e.g. destroyed spaces in Detonate the Helicarrier) you always fill a space as soon as it opens up.