I think you get dojo registration if you are lucario rank beaten all makuhita dojo areas and um uh be at level 100
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the final maze is unlocked after you beat primal dialga and complete the other mazes. You can recruit Jirachi, Suicune and Moltres in the final maze.
Articuno is in the Final Maze in Marowak's Dojo. You can also find many of the other ledgendaries there.
Card-jitsu shadow is a mystery that can somehow be unlocked in the dojo and I USED to know how to get there but after the card-jitsu party, I had no clue.
He won't come out.
You can't get the badge to the fighting gym. You can however, fight the dojo master. He is in Mt. Mortar. When you beat him, he will give you a Tyrogue.