Other mistake: Beckett points out to the murderous security guard that some graffiti magically appears on the tape, showing the live image was replaced with older pre-recorded footage. "Do you see that? It's actually kinda hard to spot." Except the graffiti doesn't appear at the point where the live feed restarts (when the guard enters the shot), it only appears on the footage after a cut away and then back.
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)
Continuity mistake: The guy at the door says "I purchased a ton of crap at that place" and the door is partially open next to him. It cuts to a barely different angle as he says "come on in" and the door is instantly wide open. (00:25:25)
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.
Continuity mistake: Castle gives Beckett a coffee mug in a very loving way by moving her hand to hold the mug by the handle while she is reading a file and not looking. Then in the very next scene she is holding the mug on the bottom in her palm. (00:05:24)
Factual error: In the beginning of the episode, the neighbors do not hear shots fired because the murderer uses a homemade silencer. At the end, when they find the murder weapon in the dumpster at the murderer's apartment, it is a revolver. Silencers (homemade or otherwise) do not work on revolvers because the gas escapes from the sides, not exclusively the end of the barrel.
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.