Other mistake: When Harry spits baby food on the baby (the camera lens in actuality) in retaliation for the baby spitting food on him and Tommy, Joseph Gordon-Levitt breaks character and laughs uncontrollably and looks at and gestures to off-screen crew members a couple times. (00:06:25)
Other mistake: When Dennis Rodman sees Harry, Tommy, and Sally at the hardware store, he addresses all three by name and asks what they are doing on Earth. If he didn't previously know they were on Earth, then he couldn't have known they were going by those names, as they chose those names after arriving at Earth. Harry even mentioned in a later episode that the names Tommy, Dick, and Harry were chosen by chance when they saw the phrase "Every Tom, Dick, and Harry" written down.
Other mistake: When Dick is trying to get into the jewelry store, the clerk keeps trying to buzz him in. When that doesn't work because Dick doesn't understand to wait for the buzzer to finish before he tries opening the door, the clerk goes over to the door and unlocks the deadbolt and opens the door for him. If the door was locked by a deadbolt, then a buzzer wouldn't really do anything to unlock the door. (00:14:05 - 00:15:00)
Other mistake: When Dick is up in the tree fort, the aliens are looking at him through their hand binoculars, but the image is taken from the same height as Dick. However, since he is higher than they are, they would be seeing him at an angle. The view as seen would be impossible unless they were at the same height as he was.
Other mistake: While talking to his therapist, Dick tells her that one of the mishaps that he caused in his relationship with Mary was showing up to a fundraiser naked. While true, this doesn't happen until Episode 13 (Rutherford Beauty) - two episodes later. This was due to the fact that most of Season 5 was aired out of production order.
Other mistake: When Dick and Sally are playing with the cat toy in the kitchen, they make a lot of joyous noise while doing so. When the shot changes, we see Mary coming up the stairs to a completely silent home, and she can't seem to locate anyone until Harry pulls her into the closet. The kitchen is only a few feet away from the stairway, so not only should Dick and Sally's voices have still been very audible, but at the very least, Sally should have been in Mary's direct line of sight. (00:15:53 - 00:16:20)