Visible crew/equipment: When Doug says he doesn't want to look after Arthur in the bedroom, there are 4 tape markers visible on the floor at the foot of the bed. (00:14:30)
Continuity mistake: When arguing with Arthur about moving upstairs, Carrie has her hand on a chair or the table depending on the shot. (00:05:50)
Continuity mistake: Arthur throws Doug's pumpkin out near the start and there is no pumpkin on the floor in the kitchen. A few minutes later in the scene, there is pumpkin on the floor in the kitchen. (00:03:40)
No Retreat - S4-E5
Continuity mistake: When Horseback riding Doug is holding on to the straps with either one or two hands depending on the shot. (00:16:50)
Continuity mistake: When Deacon is asking Doug if "Chain of causation" is a legal term, him arm goes from straight to bent between shots. (00:17:20)
Other mistake: Carrie says that Deacon doesn't drink or smoke. Many episodes have shown Deacon drinking a beer or a margarita.
Wish Boned - S7-E17
Visible crew/equipment: When Doug starts to walk over to the flight attendant, a second camera slides into the shot for a few seconds to the right of the screen. (00:12:36)
Visible crew/equipment: When the camera cuts into the wide shot of Carrie leaving the store, the camera rig to the left of the screen is almost entirely visible. You can see it follow Carrie around for a good 10 seconds. (00:10:35)
Mean Streak - S4-E3
Visible crew/equipment: After giving an interview Doug walks over to Deacon and a tape marker from earlier in the scene is visible on the floor. (00:05:20)
Continuity mistake: When Carrie is on the phone to change the insurance cover, her hair is either tucked behind the phone or covering the phone depending on the shot. (00:15:50)
Visible crew/equipment: Carrie begs Arthur to move upstairs, when she does, a tape marker is visible on the floor by the stairs to the basement. (00:05:30)
Visible crew/equipment: When Carrie walks into the bedroom at the start, 2 tape markers are visible on the green carpet. (00:06:55)
Revealing mistake: When Carrie leans on the fence to hide from the Saskeys, the fence moves. Fair enough Carrie is not a professional carpenter and has put the fence up in an afternoon. However, Tim, explains that the fence is not likely to fall due to how well Carrie has built it. (00:06:00)
Visible crew/equipment: We see Doug slide past the camera at Carrie's hospital appointment. When he does, a boom mic is reflected in the picture behind them. (00:11:50)
Visible crew/equipment: When Doug says about Kelly being in the wrong he puts his arm up in the air which shows the top of the set and some equipment. (00:14:00)
Continuity mistake: When Carrie turns round and tells Doug that she's telling Kelly about the card, the bottle top on the table is constantly changing position without being touched. (00:11:40)
Visible crew/equipment: When Doug tells Carrie "eyes off" someone can be seen moving on the stairs behind them due to a foot appearing in shot and shadows moving. However no-one else is at home other than Doug and Carrie. (00:10:40)
Continuity mistake: This is a very tricky one to spot as it happens as the shot is fading into black to a scene change but it is noticeable without the use of pause and/or slow motion. When Doug and Deacon are outside talking, Carrie comes out and then proceeds to walk back in and and audibly shut the back door. However as Doug and Deacon walk back in as the shot is fading away, the door is wide open. (00:09:30)
Audio problem: When Doug is having his photo taken the guy taking the photos doesn't actually press the button on the camera but the sounds are heard. (00:15:40)
Answer: To answer the question directly. Carrie and Doug do not have a biological child during the show's main run. In season 3 or 4 finale (cant remember which) Carrie has a miscarriage. Later on in the show they find out Carrie has 1 working ovary. Thus reducing the chance of them conceiving. By the final season, they couple agree to adopt from China and go to collect their child in the show's final episode. While on the flight back from China it's revealed that Carrie is once again pregnant. However the last scene of the show is a flash forward showing that Carrie goes full term with that child and they are struggling with both kids.
Ssiscool ★
Adding on to this answer, I read somewhere that producers were afraid of the show having a "before baby" feeling and then an "after baby" feeling. Some fans feel that main characters, or the show in general, decline in quality after a major change. For example, Jesse and Rebecca on "Full House" after they had twin sons, and Daphne on "Frasier" after she married Niles.