A K Vikhagen
Yes to all of these - you may freely add adventurers to a quest during the quest management phase, even those that have started before that round.
page 22 in rulebook 2nd ed:
Adventurers can be applied to any active quest the player has, not just the quest the player acquired that round, and an adventurer applied to a quest remains on the quest with his equipped weapon until the player removes him or the quest is failed or completed.
about the rewards, page 24:
Adventurers that were present for the completion of the quest have an opportunity to level up.