Title reference: "Limey" is American slang for an Englishman or other Briton, or someone of English descent.
Kate Beckett uses the alias "Jean Harrington" when speaking to Nigel Wyndham during the reception in the British Consulate. In "The Lady Eve (1941)", Jean Harrington is a woman trying to con a wealthy man.
The building "Scotland Yard" has not been referred to as such since 1967; since which the headquarters have moved twice with each location being called "New Scotland Yard". The name refers to the building, and is not the name of the agency, which is the Metropolitan Police Service.
Beckett senses that something is different between her and Castle. In a bit of dramatic irony, the audience knows, while Beckett does not, what that difference is: in the previous episode, he learned that for the past year, she has been lying about not remembering her shooting - including his profession of love for her - and is growing impatient waiting for her to admit and act on her own feelings.
Both the name Scotland Yard and the badge brandished by DI Hunt are inaccurate. Scotland Yard is actually the name of the building that is the headquarters of the Metropolitan Police Service, the force responsible for policing Greater London. The badge is a complete fantasy.