When Castle goes on the stand the second time, the bailiff swears him in a second time. Castle has already been sworn in, and wouldn't have to be sworn in again. The bailiff would just remind him that he was still under oath.
Martha and Alexis make their prank call from Castle's office. Once the call is known to be a fake, given the victim, the call would have been traced and the pranksters uncovered.
Castle says he needs "a listing of all the Hudson University students and alumnus in the Tri-state area." Alumnus is the singular of alumni; Rick should have used the latter form.