Castle tells Beckett being without her makes him Sherlock without Watson; Beckett corrects that to Watson without Sherlock. In the original stories it was Dr. Watson who chronicled the cases Sherlock solved, writing up their adventures for the newspapers. All the stories are from Watson's perspective and in his voice.
Castle mentions providing his ESL class with good French cuisine and Javier jokes about poutine. Poutine has become the identifying dish of Quebec, the center of Canadian Frenchness. It is french fried potatoes mixed with small cubes of cheese curd and covered with brown gravy.
When Castle goes undercover at the English as a Second Language class, he pretends to be a French Canadian. Nathan Fillion actually is a native Canadian.
Castle is elated at finally having the opportunity to say "J'Accuse". J'Accuse (I accuse (you)), became a famous exclamation during the Dreyfus Affair in France in 1898. An open letter published on the front page of a Paris newspaper was headlined "J'Accuse...!" and condemned the French president and government for gross antisemitism. The phrase has become associated with holding high and powerful folks accountable.
It is revealed that Dr. Perlmutter has an identical twin named Edgar.