If I clear a room of monsters and have a healer, can I heal my group before moving to next part of the quest? Or take other actions for that matter?
If you're playing Road to Legend and you've successfully killed all the monsters, you can spend as many turns as you like taking rest actions and using special abilities to heal and running back for any search tokens you passed.
BUT you'll likely die if you do this. Road to Legend's "Peril" system means that the more turns you spend in a dungeon, the more likely you are to suffer highly damaging peril effects. If you spend 2-3 turns healing after ever room, by the time you reach the end of the dungeon you'll be suffering huge amounts of damage from peril, or having additional monster groups hurled at you, or both.
Descent is a race--even in the Road to Legend form. Every turn you spend NOT moving directly towards your goal will make it that much harder to succeed in the end.