When Beckett calls Castle, at the beginning of the episode, her picture ID is the cover of Heat Rises, the third Nikki Heat book.
Twice Castle tells Alexis that they could become the father-daughter version of Grey Gardens. 'Grey Gardens' was a 1975 documentary about an 80 year old woman and her middle aged daughter. Formerly rich members of upper society they were now poor, socially isolated, possibly crazed, and living in the squalid remains of their rotting family mansion. All they had was each other and their dreams of past glory.
Castle tells Victor Baron, "Looks like you were hoisted on your own Picard," referring to the painting, and a play on "Hoist with his own petard," a phrase meaning, "Defeated by your own weapon," that William Shakespeare coined in "Hamlet."
Victor Baron is clearly based on Donald Trump, and his last name is a slight misspelled version of the name of Trump's youngest son. Barron.
Bobby Stark's (Sasha Roiz) character, and name, are a clear play on Bob Saget, including the references to "America's Dad".