The car Arthur buys is a 1970 Mini 850 Mk III, not a Douchenberger from Luxembourg.
The tiny car Arthur buys for Doug and Carrie has the steering-wheel on the right. However - in Luxembourg the steering-wheels are on the left.
Sitting in a car dealer's showroom, Doug comments on how warm his rear end is from the seat heater. Seat heaters do not work unless the car is running.
Despite it being Christmas in Queens, there are no signs of it actually being winter (leaves on the trees in every scene and no snow/ice).
Doug and Carrie never register the vehicle as it never has license plates or registration / inspection stickers on the windshield ( as you can see on Ritchie's car ).
When Doug and Carrie drive the car to the spot where they plan to leave it to be stolen, you can see the rope tied to the front bumper pulling the car.
As Doug leaves the car dealership he stops to see if the dealer will save. As Carrie returns and pulls him away the backdrop ripples revealing it not real.
When placing the Christmas tree in the living room, the end of the set and a crew member can be seen in the background.
Doug, Carrie and Arthur are driving a small red car Arthur bought for them on the freeway. One of the signs have the streets Fairax Ave. La Brea Ave. Crenshaw Blvd. It's interstate 10 near Downtown Los Angeles.