Character mistake: Castle claims that they are technically standing on Russian soil inside the consulate. Contrary to popular belief foreign diplomatic missions, while being afforded special privileges (ie. Diplomatic immunity), are not considered sovereign territory of the represented state. Because of that Castle and Beckett are in fact on U.S. soil.
Backstabber - S8-E18
Continuity mistake: When Wesley changes his statement after Hayley gives him a "swirly", his wet hair changes position between shots.
Audio problem: When Martha arrives at the beginning of the episode, she says to Castle, "I didn't want to disturb you in case you wanted to sleep in." But her lips don't move when she says "sleep in."
Death Wish - S8-E17
Continuity mistake: Beckett keeps a jar of nuts on her desk. When she's talking to Castle about Lars, she reaches over her desk to put the jar down. But in the following shot when Castle stands up, both her hands are suddenly in front of her laptop, clasped together.
Death Wish - S8-E17
Continuity mistake: When Castle beings theorising that Genevieve might be a genie and that she wants him to be her new master, Beckett is holding a glass in both hands. In the following shot, she's holding it with just one hand.
Continuity mistake: When Castle gives Beckett a mug of coffee at the beginning of the episode, she isn't holding it by the handle, but in the shot immediately afterwards she is.
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.