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)
Continuity mistake: When Doug spots the chicken he has his gun down by his side in the first shot and raised in the second shot. (00:18:40)
Continuity mistake: When Doug goes to get Carrie's present from the closet her position on the bed changes between shots. (00:05:30)
Hi Def-Jam - S3-E13
Continuity mistake: At the start Carrie is holding an egg in two hands or one hand depending on the camera angle. (00:00:42)
Strike Too - S3-E6
Continuity mistake: At the start when Doug is having his breakfast something appears on top of his pancakes. (00:00:30)
Visible crew/equipment: When Fisker comes down to the drivers area of IPS, there is a white T Mark on the floor. (00:14:20)
Strike Too - S3-E6
Continuity mistake: When Carrie puts her bag on at the start it swaps arms instantly. (00:02:30)
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.