Phoenix put the hat back onto Shadi's head because when he talked with Kristoph, Kristoph made a joke about Shadi being bald. By putting the hat back on Shadi's head, this proves that Kristoph had to have seen Shadi's bald head at the time of the murder, rather than after, therefore proving he was there the moment of the murder.

As for the ace, Apollo didn't get in trouble because he didn't get caught. The only people that knew it was a forgery were Phoenix and Kristoph, and neither of them complained.