What if the player who took the tile for Mah Jongg made a mistake, and now the hand is dead. Does the other claimant get to take the tile now?

It depends. If both players wanted the tile for Mah Jongg, then yes the second claimant now gets the tile. But, if the second claimant wanted the tile for an exposure they do not get it. In this case, the tile stays with the dead hand.