Audio problem: The animated Christmas movie the boys are making is really "the Spirit of Christmas", the movie Trey Parker and Matt Stone made before South Park. The dialogue recorded by the boys is the same as in the original (with a few changes, but when the film is shown later it's the original speech from "The Spirit of Christmas". You can tell by the way the voices are done, and the tone of the voices.
Continuity mistake: When Kyle tells Mr. Hankey, Autumn, Cornwallis, Amber and Simon about the $300 he, Cartman, Stan and Kenny got for the animated Christmas card, the olive stick in Autumn's martini glass is facing to the right. But when Autumn says to Mr. Hankey "That's not the only thing we've got to get working again if you know what I mean", the olive stick in the martini glass is now facing to the left. The olive stick then returns to facing to the right for the remainder of the scene.