The name on Jerry Tyson's (3XK) badge when he threatens Castle in the precinct is J. Rook. In Castle's Nikki Heat books, Jameson Rook is the character Castle based on himself.
Castle identifies "Joseph Vacher", an alias Jerry Tyson used during his framing of Castle, as a serial killer from 19th century France. Active from 1894 to 1897, Vacher murdered and mutilated at least eleven people, and was often compared to Jack the Ripper.
In an early scene at the precinct, when Castle offers Beckett a cup of coffee, she is too distracted by the file she's reading to accept it, so he has to physically take her hand and put it around the cup handle.