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 Arthur walks in near the start, Carrie is looking at him. Camera cuts and she is now looking in the totally opposite direction.
Revealing mistake: Just as Arthur goes to look in Carrie bag, there is a wide shot and a cabinet can be seen by the door. In the glass of this cabinet's door, you can see a stage light reflected. (00:11:45)
Continuity mistake: After Doug has shot himself, the position of the staple gun on the bench keeps changing without anyone touching it. (00:06:40)
Continuity mistake: After trying the Sgt. Salty cereal, while talking to Carrie, Doug gets out a plate to make something else to eat. After he puts the plate down on the table, the cereal box moves and switches sides. (00:13:30)
Other mistake: When Doug is drawing a picture to explain to the nurse why he is there, a man purchases a beverage from the "Hot Drinks" machine behind Doug. The drink is ready unrealistically fast and the man immediately takes a sip from an obviously empty cup. Those machines produce notoriously hot beverages, so if the machine actually worked the man would have scalded himself.
Big Dougie - S2-E21
Revealing mistake: When Doug is jogging in the marathon he is casting three shadows in different directions. Also, the female jogger behind him appears to be in good shape but is still jogging as dramatically slow as Doug. (00:19:20)
Deliberate mistake: At the beginning of the episode, when Doug and Carrie are in the shower, you see their silhouettes. The first mistake is that for the silhouettes to show like that they have to be evenly backlit which wouldn't make sense in the shower. The second mistake is a little more subtle; you can see the outline where the actor's bathing suits are (inevitably) slightly larger than the bodies. There is also a slight darkening in the areas where the bathing suits cover skin. (both of these things are much easier to see on Carrie than Doug because of her smoother figure and skin tone).
Soft Touch - S2-E22
Continuity mistake: When Doug is getting into bed while talking to Carrie about selling water filters, his grip on the duvet changes instantly. (00:10:00)
Visible crew/equipment: When the group enters the house at the start, there are a few little white dots strategically placed on the floor where actors stop. Clearly place markers. (00:02:00)
Deliberate mistake: When Arthur enters the living room playing the harmonica, it is obvious he is not actually playing it. (01:23:10)
Revealing mistake: When the lights go out at the end of the episode, there is a reflection from a stage light very visible. (00:20:30)
Party Favor - S2-E13
Continuity mistake: When Doug is reading his speech to Arthur, Arthur's legs are crossed or uncrossed depending on the angle. (00:09:30)
Soft Touch - S2-E22
Continuity mistake: When Doug is telling Arthur about the filters, there is a leaflet on the table. Between camera cuts, this leaflet is constantly changing position. (00:13:25)
Visible crew/equipment: When Doug walks out of Carrie's room, a boom shadow can be seen just outside the room on the wall. (00:03:35)
Queasy Rider - S2-E1
Continuity mistake: When Carrie says to Doug that he can't have a bike, her hand are behind her back. Camera cuts and now her bag is on her shoulder and her hands are on her bag. (00:03:00)
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.