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.
Nanny McDead - S1-E2
Factual error: The establishing shot of the building where the murder takes place features on the foyer quite prominently "715 - The Langham." The story takes place in New York, but The Langham (named after the famous British hotel) is the name of a residential building in 715 S Normandie Avenue in Koreatown, Los Angeles, used in this episode. There is a hotel by that name in New York, not an apartment building, on 5th Avenue. (00:01:00)
Flowers for Your Grave - S1-E1
Other mistake: Castle's book "Storm Fall" is a bit of an enigma, since it is amongst all the other books in the collection of the scapegoat killer when Kate opens it, but Castle gifts "an advance copy" to her later, while in the very first scene of the show there is a party with the book already in the hands of the public. Reviews and a proper release appear only in the fourth episode.
Flowers for Your Grave - S1-E1
Character mistake: At the end of the episode, Castle finds his inspiration again and there's a word processor page fading in/out with text being written. Funnily enough, whoever typed in overlooked something, since there's a sentence that says "Begfore she felt safe." (00:41:30)
Factual error: The con is said to be electrocuted. But the jump start cables are connected to handcuffs which would short the electrical circuit and most probably melt the handcuff chain, but wouldn't electrocute the victim. (00:02:50)
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.