Deliberate mistake: At the beginning of the episode, when Doug and Carrie are in the shower, you see their silhouettes. The first mistake is that for the silhouettes to show like that they have to be evenly backlit which wouldn't make sense in the shower. The second mistake is a little more subtle; you can see the outline where the actor's bathing suits are (inevitably) slightly larger than the bodies. There is also a slight darkening in the areas where the bathing suits cover skin. (both of these things are much easier to see on Carrie than Doug because of her smoother figure and skin tone).
Deliberate mistake: When Arthur enters the living room playing the harmonica, it is obvious he is not actually playing it. (01:23:10)
Answer: Sara hasn't been mentioned in later episodes, and it suggested in later episodes that she never existed, and that Carrie is an only child. Kevin James stated that the producers couldn't think of an good plots to use with Sara, so they left her out.