Hypothetically, if one has a Veteran Sellswords on a quest card, and then that Sellswords flips to an opponent's quest zone, and then comes back to yours, can that card go back onto the quest? Or does that Sellsword now have to not be questing?

More generally, when a card moves into a QZ, can it be put on a quest in that quest zone?

Nope. Page 11 , "To send a unit on a quest, a player plays the unit on top of the quest card when he plays that unit from his hand"