Carcassonne fairy - scoring with a fairy for multiple areas and at the end
As per the rules:


  When a feature (city, road, cloister, or farm) with the fairy is scored, the player whose follower stands on the tile with the fairy scores 3 points for standing there, and may score for the feature as normal. When scored, the follower returns to its owner, but the fairy remains. 


Make special note of "When scored, the follower returns to its owner, but the fairy remains."
This means that you can only get the 3 bonus points by scoring something which will remove your follower from the tile. This can lead to multiple bonuses. The fairy is placed at the time you would place a follower, which comes before scoring (this should answer number 2).

As for the monastery, once again the rules state that you should return your meeple, and I don't see that happening with the monastery being finished (unless another feature is finished along the way).