1) Dojo Registration is a hidden dungeon. This dungeon, as I remember, has 50 floors, with some weak Pokemon. I think its purpose is to register for training at Makuhita Dojo at first, but then it doesn't make sense so it is hidden.
2) Deoxys will go with its Friend Area when you recruit it. But you must be at least Lv60 (w/o Friend Bow) or Lv30 (w/ Friend Bow) to recruit it due to its negative percentage of -10%.
3) Nothing, only a statue of Lucario standing in front of your base.