The director is impressed by Monk memorizing the entire play in just one viewing. But Monk's first viewing of the play (he asked Sharona when her sister was going to come on stage, so he obviously had not seen it) was interrupted by the murder. However, he wouldn't have to know the entire play. He was playing a character, Burt, who is murdered before the play is interrupted.
In the flashback when Monk is explaining how the crime was committed, the supposed doctor takes a knife from his jacket and buries it in the actor's chest. The first thing the police would have done is check the knife for prints. This would have solved the case immediately because Gail's prints would never have been found on the murder weapon. However, this is addressed when Monk visits Gail in prison. He explains that the police found no usable prints on the weapon and that they are assuming that Gail wiped them off during all the confusion.
When the actor is stabbed onstage, the knife sticks out of his chest on its own, even though his costar used a prop blade that did not penetrate him.
Gail was seen holding the knife that stabbed him, but the actual weapon didn't have her fingerprints.
When Monk pulls the 3 dollar bills out of his wallet to bribe the doorman the bills are visibly creased. The wallet is also different from "Mr. Monk Takes a Vacation (2002)". Monk isn't exactly a person who would have more than one wallet. For example, he bought three identical lamps so he'd have backups.