Continuity mistake: Carrie asks Doug what he's doing when she walks into the bedroom. She then hangs her coat on a hanger and has the coat in her hands (as seen in her reflection in the mirror). Camera changes and she's shutting the wardrobe. (00:08:01)
Continuity mistake: When Carrie suggests about them looking after the kids to Doug, he has his arm up by his chest. When the camera changes it's lowered. (00:04:20)
Strike Too - S3-E6
Continuity mistake: When Stephanie is placing cones, she is looking at Doug with a concerned look upon her face. When the camera changes she's now facing away and laughing. (00:16:20)
Character mistake: In this season we see Doug smoking cigars and cigarettes. However in S2 E1 Queasy Rider, Doug is shown to absolutely detest the smell and taste of smoke from these. It is highly unlikely that he would then take up smoking.
Strike One - S3-E5
Continuity mistake: When Carrie says about smelling like the F train she has her hands together in front of her chest. When the camera changes angle, her hands are spread. (00:16:20)
Strike One - S3-E5
Visible crew/equipment: When Doug is listing all the extras for the truck, you can see tape markers on the floor behind them. Specifically, one in between the feet of an extra that is standing behind them. (00:08:50)
Continuity mistake: When Doug is writing on a board, he holds the board at the end or the top depending on the shot. (00:03:30)
Continuity mistake: After Carrie says she's taking Doug's camera to her office party, the box magically appears on her lap. (00:09:50)
Continuity mistake: When talking in the bedroom, the magazines on the bed change from on the bed to on Carrie's ankles. (00:07:20)
Strike Out - S3-E7
Continuity mistake: The bottle of Lipton Ice Tea on the table rotates on its own without anyone touching it. (00:03:20)
Visible crew/equipment: When Doug is outside talking to Deacon and Arthur opens the kitchen door, as the door swings back you can see all the set lights reflected in the glass. (00:08:20)
Continuity mistake: When Kelly says to Deacon "Maybe it's 'cause you're walking late all the time" she has her mug up in the air, camera cuts and it's down on the table. (00:06:25)
Continuity mistake: When Arthur walks into the kitchen at the start and picks up the phone, the door is closing. The camera cuts to Doug at IPS, then back to Arthur and the door is still at the same point of closing. (00:00:55)
Visible crew/equipment: When Doug sneaks out of Carrie's house and finds her having a cigarette by the car, a boom mic shadow can be seen on the bush beside Doug. (02:53:45)
Continuity mistake: When Carrie is sitting at the table talking to Doug about not finishing school, her hand is beside the text book and then instantly holding it from underneath in the next shot. (00:10:35)
Continuity mistake: When Carrie is having lunch with the lawyers from her office, she is holding a cup and then instantly holding a napkin in the next shot. The cup and napkin switch again depending on the shot. (00:03:10)
Continuity mistake: When Doug and Carrie are arguing about not having gone out for pizza, Carrie has her pen in her hand, but when it cuts it's now on the desk. (00:10:30)
Continuity mistake: When Doug and Carrie are in Deacon's kids' room, Carrie is changing the baby and between shots the baby changes from lying down to sitting up. (00:15:00)
Inner Tube - S3-E17
Continuity mistake: At the start at IPS, Doug's tie is looped over itself twice. Camera cuts and now it's not looped at all. (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.