Revealing mistake: Throughout the cockpit scenes, Otto's (the auto pilot) face is just painted, with no facial features sticking out. When he is about to fly the plane, he winks out the window, and turns his head. After he turns his head you can see the extra attachment on his left eye used for this. (01:22:40)
Revealing mistake: When the doctor is removing the eggs from the woman's mouth, he actually has them in his hand, not taking them out of her mouth. A simple sleight of hand, but still noticeable.
Answer: This is a reference to The Godfather, where a character being intimidated by the Mafia wakes up with his prize racehorse's severed head next to him on the bed. For comedy purposes this is twisted by the movie as an implied sexual relationship, when the horse is revealed to be alive.