When describing the jade elephants, they are said to be precious artifacts from China; they even reference the Forbidden City, which is in China. For the rest of the episode, however, they are dealing with members of the Japanese government.
When Peter and Neal are arguing in the street, after Alex shows up at the meet and the suspect has gotten away, Neal's coat collar is popped in the shots taken from behind him and lying flat when the camera's in front of him.
Jade elephants were originally gift of a Chinese Emperor, but the Japanese ambassador wants them returned.
When Peter moves the chess piece at Neil's place, in the long shots the white pieces are to the right of the screen but in close up the black pieces are.
During the search of Peter's house, Neal primes various audio-visual equipment to come on when the power is restored. Most of Peter's equipment has touch controls or micro-switches which don't respond without power; Neal's trick would only have worked with mechanical switches.
The judge is federal, specializing in probate. Probate is almost exclusively a matter for state courts. A federal judge cannot dispose of property under state probate jurisdiction, including such things as a foreclosed house. It is unlikely if not impossible that enough probate-related cases would rise to the level of federal jurisdiction to allow her to specialize in them, and federal cases in New York are assigned randomly in the first place.
When Neil looks at Pierce Spelman's driver's license, the card says British Columbia although the writing is in French and it is actually a Quebec driver's license.
Incorrectly regarded as error. The Chinese emperor gave expensive gifts to visiting dignitaries. The elephants were given to the Japanese. The Japanese ambassador correctly wants them returned.