Visible crew/equipment: When Castle pokes his head through the doorway with his eyes closed before walking in on Alexis again, in the shot facing Alexis the boom mic hovers in front of her, at the bottom of the screen. (00:40:50)
Character mistake: They find the DeLorean and the licence plate is shown: LFM-26W1. Esposito calls in and says the car needs towing: "'81-'82 DeLorean, New York plates, JLD..." Nothing like the actual licence plate.
Answer: The networks are now having winter breaks. A mid season end of new episodes, when they come back after the new year, it picks up as if several months have gone by. Beckett suggested Valentine's Day as a "chick flick" to watch as she is moved in. Also some episodes are not scheduled to air in order.