In the opening scene, the female character picks up some mail from the floor that included a couple of magazines. In the next shot, all the woman has in her hand are envelopes. The magazines are gone.
Lanie determines that Tessa's killer was wearing gloves because of talcum powder found on Tessa's body, and likens them to the gloves Beckett and company use at crime scenes. CSI/CSUs don't use talcum powder in their gloves, because of the potential for cross-contamination of the scene.
When Castle & Beckett are talking in the precinct, as Castle hands Kate a cup of coffee, it is clear she is holding the mug by the handle. As soon as the camera changes, she is seen holding it by the bottom.
When they do the close-up of the fingerprint chart from CSU, one fingerprint is identified as being on the "door jam". The part of a door frame is a door jamb.