Now that I have actually played the game, I have a couple of questions just to be sure I understand correctly:

1. Does Hex activate once per run and then gets turned down just like the Hunter Active Abilities?

2. When Hazardous triggers, does the top card get discarded before or after you draw to replace playing the move card? (I have been drawing to replace and then discarding, reason being that I see the draw as a part of resolving the movement).

Yes

Yes, that “After a Move” means after that Move resolves, so you’ve been playing it correctly. “During a Move” or “When a Move” would occur as part of that action.