Just after Kate tells Richard to stop joking, there is a clear view of the cuff on her wrist; it is so loose that she should easily be able to free herself. Shots before and after this point show the cuff to be closed properly.
When Beckett is on Castle's shoulders trying to get the hatch open, her arm is very clearly not cuffed to his in the overhead shot.
When Beckett first attempts to climb onto Casle's shoulders, he starts giving her a hand-boost twice.
Lanie states that the victim's fingerprint comes from the "epidural layer" of skin. Instead, it should be "epidermal."