To obtain reluctant witnesses' cooperation, Castle will occasionally offer to name a character in his Nikki Heat books after them, often right in front of Beckett, or Ryan and Esposito. While it might appear to be a harmless offer, any competent defense attorney could paint it as bribery and witness tampering, which are both illegal; as cops, they don't seem bothered by an action that could potentially get a solid case against a suspect thrown out at trial.
In many episodes detectives track somebody by using the entrance/exit information from their MetroCards. In NYC the card is only swept once at the entrance but not on the exit (except on Staten Island where card is swept at the exit only), so the person can travel almost the entire subway system with only the entrance recorded. Exit swipes are in use in Washington, DC and San Francisco's BART where zonal fares are in use.
We see the Castle constantly ask the detectives working with him (or the detectives themselves) to put out APB's (All Points Bulletins) or BOLOs (Be On the Look Out) to bring suspects and witnesses in. When found NYPD just walk up to people and place them in the squad car, taking them to the station for the Detectives and Castle to interview even though it's pretty clear they didn't want to come. If there is no probable cause for an arrest or an active arrest warrant, the police can't make you go anywhere against your will. Once in a while, a wealthy or educated person will assert this and ask a lawyer to be present but this is rare.
Throughout the show Beckett goes on undercover operations from time to time. However, with all of the publicity associated with the Nikki Heat books that are based on her, it is more than likely her "cover" would be blown and would be useless in that role.
The goof items below may give away important plot points.
During the storyline arc of their separation in season eight, Castle fears he's lost Beckett's love and spends a great deal of time and effort trying to win her back. While not entirely unjustified, his fear is a little over-exaggerated. On numerous occasions, more than simply telling him, Beckett has demonstrated how much she truly loves him; perhaps most notably, she is willing to risk not only exposing their relationship to Captain Gates, but losing her job in the process, to comfort him when Alexis is abducted in season five.