When Beckett is informed that Castle wants to help her work the case she asks Captain Montgomery if she could talk to him for a moment privately. In the script, Montgomery was supposed to say yes, but instead he said no and shut the office door. Kate's reaction is the honest reaction as the director liked Stana's shocked reaction so much they kept it.
Because this was the pilot, it was filmed in two locations: New York and L.A. This is most clearly evidenced during the precinct scenes. There are obviously two different precincts; one is the precinct used in all other episodes (as of 5x17) and the other shows Montgomery in a completely different office, as well as a nice red, brick wall in the background behind Beckett's desk. Also proof of the 2 sets is Beckett's desk location, as well as the items on her desk and their locations.
When Castle is guessing the "story" of why Beckett became a cop, he deduces that she's a Manhattanite because she's "not bridge and tunnel". 'Bridge and tunnel' is a phrase used in Manhattan that refers to the fact that to reach the city, citizens from the other boroughs have to commute over one of the numerous bridges into the city, or through the Lincoln and Holland Tunnels that run under the Hudson River.
Castle is thrilled about having a copycat, calling the dubious honor "the criminal Cooperstown". Cooperstown is a village in New York that is home to the Baseball Hall of Fame, and its name is often used as a metonym to refer to the Hall of Fame itself.
Keir Dullea plays the father of the episode's murder victim. Castle (2009) later paid homage to 2001: A Space Odyssey (1968), the film for which Dullea is most famous, in The Wrong Stuff (2015).