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)
Strike One - S3-E5
Visible crew/equipment: When we see the car in the garage for the first time we see the camera start to move. As it does, so a boom shadow can be seen to move over the car. (00:09:20)
Continuity mistake: When Doug and Deacon both volunteer for the calendar the magazine on the bench is constantly changing position without being touched. (00:05:10)
Continuity mistake: When Carrie is unpacking the shopping at the start she holds up an empty bag that magically folds itself flat between camera cuts. (00:01:50)
Strike One - S3-E5
Other mistake: The salesman calls Doug by his name without Doug telling him his name. His Work ID as shown in other episodes states D. Heffernan. (00:07:20)
Visible crew/equipment: When Doug arrives at the airport and we see the two gentlemen writing names on a board, everyone has tape markers at their feet. (00:03:20)
Visible crew/equipment: When Doug and Arthur arrive in the City to talk to Carrie, yellow tape markers are visible on the floor by Doug's feet. (00:15:10)
Visible crew/equipment: When Carrie comes back saying she can't swap the flights, there are various tape markers visible on the floor. (00:19:00)
Continuity mistake: When Carrie realises Doug has been in the pool, the magazine in Doug's hand vanishes instantly. (00:18:20)
Visible crew/equipment: When we first see Doug on a bike in an overhead shot the camera rig is reflected in a car windscreen. (00:17:50)
Visible crew/equipment: When Carrie gives Doug the card, the top of the set is visible behind them. (00:10:40)
Visible crew/equipment: In the garage, when Spence says "so Deacon" there is a shot of Danny in the chair with a taper marker in front of him. (00:06:30)
Visible crew/equipment: When Deacon shows up with no shoes he steps inside the house and there is a white tape marker visible where he stops. (00:15:20)
Visible crew/equipment: When Doug says about Carrie's laugh he sits down on the sofa. As he does, part of the set lights are visible above him. (00:15:20)
Continuity mistake: When talking in the bedroom, Carrie's legs are constantly alternating between on the floor and crossed on the bed. (00:07:20)
Visible crew/equipment: During Spence and Danny's fight scene, there are a couple of shots where you can see Danny's mic pack on his belt. (00:16:20)
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.