What can you do with challenges after completing them in an area?
Challenges reset every day. Once you've done them for the day, that's it - you can't complete them again. When you complete them, they typically reward you a token that gives rep.
Basically every activity in the game (going to each planet, strikes, Crucible playlists, etc.) has their own set to challenges. There is also a daily milestone for completing challenges that resets daily ("Complete 5 Challenges", "Complete 3 Strike Challenges", etc.) that rewards a reputation bundle.
If you played the original Destiny, they seem to be Destiny 2's version of bounties.