Once an enemy engages a player. If it survives to the end of the round is it still considered engaged or go back to the staging area. Also do engaged monsters threat strength count in the quest phase?

Once engaged an enemy remains engaged unless put back in the staging area or switched to engage another player by card effects.

Once engaged, an enemy's threat is no longer counted during questing. Only the threat in the staging area has an impact on questing. Think of the staging area as enemies and locations that are yet to be explored/encountered and are lurking threats.