The type of aircraft changes (in the exterior view), multiple times during the flight.
The airplane that is shown taking off is a Boeing 767, which is also intermittently shown in flight. The other shots of aircraft in flight is a Boeing 737.
When Monk and Sharona arrive at Newark Airport, they are told that "Newark P.D." had been notified to make the arrest of their suspect. The Port Authority of New York and New Jersey Police have jurisdiction at the airport, not the local P.D.
Airline crew members are trained to never declare someone dead. Not only does this require a medical professional, but upon arrival, that declaration would subject the plane and every occupant to search and investigation for possible foul play. The preferred term is "unresponsive."
Adrian identifies Stefan Chabrol as someone who knows about airplanes because Stefan lifts his glass just before the airplane turns. Lifting the glass is nothing but a coincidence. Airplanes turn very gently, and the banking is to keep the apparent force of gravity directed "down." A glass left on the tray before and during a turn would display no difference compared with a glass left on a tray during level flight.
When Monk is in the lavatory he puts the lid down on the toilet, yet the bowl interior remains lit. Toilet bowls do not have interior illumination.
When the passengers on the plane to Paris received the announcement that they would be returning to the gate, they had already been served drinks. That wouldn't happen until the plane was in the air, as had already been established on the flight to Newark.
The flight attendant says it's illegal to ring the call button unless it's an emergency. This is incorrect, as it's to be used for any reason one needs the attendant. But misusing it as Monk does would be a violation of federal regulations.
The attendant was probably lying to him, to get him to stop using the button, because she was clearly exasperated with him by this point.
Near the end after the murderers are arrested and are being led out of the airport by the police, the path they walk through runs through a metal detector. The metal detector does not make any sound even though they are wearing handcuffs.
29 minutes in, when Monk is talking to Disher, the salesman next to him (played by Garry Marshall) is wearing a black sleeping mask. Seconds later and from a different camera, the sleeping mask is gone - presumably so he can exchange a few words with Monk while he's half asleep.
[33:49 ] Leigh, the flight attendant, drinks a small bottle of liquor while in flight. Required crew members, including flight attendants, are prohibited from drinking eight hours before and during a flight. Her actions would be a violation of the Federal Aviation Regulations.
When Beach says he recommends the Anaconda extension cord for retrieving the shoes, it cuts to a different angle halfway through the word "Anaconda" and Beach's mouth is closed.