Continuity mistake: In the overhead shot when Carrie parks there is no-one around. From the internal shot, there are 2 people visible through the side windows. (00:00:40)
Queasy Rider - S2-E1
Revealing mistake: When Doug comes in at the start and says he has been riding Ritchie's hog, all the stage lights can be seen in Carrie's wine glass. (00:02:20)
Continuity mistake: When Stephanie arrives and Doug says about killing his parents, Stephanie has her hands apart, but when the camera angle changes to a different view, her hands are clasped together. (00:14:40)
Revealing mistake: When we see the montage of Doug working as a manager, in one overhead shot you can see the image on the PC is simply stuck on to make it seem like it's on. (00:08:50)
Visible crew/equipment: While Doug and Carrie are in the kitchen talking about Doug's job, right after Carrie says, "Wait him out 5 or 10 years," in the next shot facing the sink, we can see the end of the kitchen set at the right side of the screen. (00:14:00)
Strike One - S3-E5
Visible crew/equipment: After Carrie discovers she can't take a shower because Doug ripped out the shower tiles, when she walks out of the bathroom keep your eye on the window at the right. Just as Doug lifts the hammer a crew member is visible in the bathroom's window (it's not Doug's reflection). (00:17:15)
Continuity mistake: When Carrie is on about having never spent that much on a vacation, she has a leaflet in her hand that is open. Camera cuts and now it is partially closed. (00:03:10)
Continuity mistake: When Doug asks Carrie to take a photo of him on the bed, there is nothing on the bed. After she says "stop", 2 items have appeared on the bed that were not in the previous shot. (00:08:45)
Continuity mistake: The leaflet in Carrie's hand at the start vanishes instantly. (00:03:10)
Continuity mistake: After throwing the dummy out the door, Doug kicks his bag out the door too, only for it to reappear a few minutes later. (00:18:20)
Continuity mistake: Doug throws the dummy out the front door and in doing so, knocks a green Jets jacket to the floor. In the next shot the jacket is back on the stand. (00:18:40)
Continuity mistake: At the buffet, the cup that is on the table is either on or off a saucer depending on the shot. (00:06:55)
Continuity mistake: When Doug and Arthur sit down to eat, the cup that is on the table is either by Arthur or by Doug depending on the shot. (00:06:35)
Continuity mistake: When Doug goes to meet up with Ray, Richie zips up his jacket. In the next shot his jacket is unzipped. (00:06:20)
Continuity mistake: Carrie and Arthur are talking about Carrie making a Thanksgiving meal, and as they are talking the frozen meals in Carrie's trolley change position between shots. (00:03:10)
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.