During a break from rehearsals in Toronto, Luke Mably (Eddie) and 'Ben Miller' (Eddie's valet, Soren) went out while in character to the Four Seasons and convinced everyone that Mably was the Prince of Denmark. They were demanding tea and cake and had six waiters around them, bowing.
During filming, Luke Mably accidentally cut himself on the meat slicer. In the shot of Paige and Eddie dancing at the coronation ball, you can see his left index finger wrapped up (in a condom, according to Luke Mably in an interview) to cover the wound.
The cabdriver, who takes Paige to the royal parade, when she has just arrived in Copenhagen is a Danish actor, Jesper Ansholt. He shouts in Danish out of the window: "Få så røven med jer deroppe!" = "Move your asses up there!"
The castle shown to be the royal palace is Frederiksborg castle in Hillerød, north of Copenhagen. It's not the royal residence, but a museum. you actually get a small tour of Copenhagen and see among other places Amalienborg Palace, which IS the residence of Queen Margrethe II of Denmark and her family.
The film includes footage from To Catch a Thief (1955), featuring Grace Kelly. Like the Julia Stiles character in this film, Kelly was also an American who became a European princess, when she married the prince of Monaco.
The bishop is a Danish actor, Claus Bue. He says in Danish: "Edvard Valdemar, du skal fra denne dag være konge. Modtag disse regalier og hav altid Danmarks bedste for øje og i dit hjerte. Modtag rigets æble, der hjælper dig til at blive den konge, Danmark fortjener." = "Edvard Valdemar, you shall from this day forward be King. Receive these regals - and always focus on the best for Denmark. Keep that in your heart. Receive the apple of the Kingdom, which will help you become the king, Denmark deserves."
The trivia item below may give away important plot points.
In the scene where Paige breaks up with Edward during his coronation in the Danish palace, there is a portrait of George Washington on the wall. Washington is a direct descendant of Halfdan, King of Denmark and Sweden.