Wild Cards - S2-E20
Factual error: Deacon is holding an In-N-Out drink cup. In-N-Out is a burger chain only located on the West Coast of the United States. (00:02:00)
Revealing mistake: When Doug and the rest are at a ballgame, the man sitting behind them is eating a hot dog. Actually he's pretending to eat, it never shows any bite marks and the guy is pretending to chew it.

Continuity mistake: When arguing about who is going to die first, the amount of whipped cream around Doug's mouth changes instantly. (00:19:20)

Continuity mistake: When talking to Deacon outside, Deacon has his arms folded or straight depending on the camera angle. This is before anyone comes out. (00:08:05)

Hi Def-Jam - S3-E13
Continuity mistake: When Doug is seeing the HD TV for the first time, he's holding his PDA in two hands which then becomes one hand when the camera has changed to a different angle. (00:02:20)

Continuity mistake: When Carrie gets her camera it changes position in her hands before she turns it around. Note the strap, it goes from left to right. (00:06:00)

Continuity mistake: Carrie asks Doug what he's doing when she walks into the bedroom. She then hangs her coat on a hanger and has the coat in her hands (as seen in her reflection in the mirror). Camera changes and she's shutting the wardrobe. (00:08:01)

Continuity mistake: When Carrie suggests about them looking after the kids to Doug, he has his arm up by his chest. When the camera changes it's lowered. (00:04:20)

Strike Too - S3-E6
Continuity mistake: When Stephanie is placing cones, she is looking at Doug with a concerned look upon her face. When the camera changes she's now facing away and laughing. (00:16:20)
Character mistake: In this season we see Doug smoking cigars and cigarettes. However in S2 E1 Queasy Rider, Doug is shown to absolutely detest the smell and taste of smoke from these. It is highly unlikely that he would then take up smoking.

Strike One - S3-E5
Continuity mistake: When Carrie says about smelling like the F train she has her hands together in front of her chest. When the camera changes angle, her hands are spread. (00:16:20)

Strike One - S3-E5
Visible crew/equipment: When Doug is listing all the extras for the truck, you can see tape markers on the floor behind them. Specifically, one in between the feet of an extra that is standing behind them. (00:08:50)

Continuity mistake: When Doug is writing on a board, he holds the board at the end or the top depending on the shot. (00:03:30)

Continuity mistake: After Carrie says she's taking Doug's camera to her office party, the box magically appears on her lap. (00:09:50)

Continuity mistake: When talking in the bedroom, the magazines on the bed change from on the bed to on Carrie's ankles. (00:07:20)

Strike Out - S3-E7
Continuity mistake: The bottle of Lipton Ice Tea on the table rotates on its own without anyone touching it. (00:03:20)
Continuity mistake: Near the end when the Heffernans, Palmers, and "Shmenkmans" are having dinner and are in conversation, Carrie is shown sitting extremely close to Abby, who is on Carrie's left. But in shots in the middle of the conversation, there is significant empty space to Carrie's left.
Dire Strayts - S2-E8
Continuity mistake: When the women come into the bedroom, Ray's hand goes from flat to raised on the bed. (00:13:20)

Continuity mistake: When Doug and Arthur are sat at the dinner table, Arthur's hands go from on the table to clasped between shots. (00:07:45)

Maybe Baby - S1-E25
Continuity mistake: When Doug sits down with Arthur in the night, the magazine on the table changes between shots. (00:11:50)
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.