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.
Revealing mistake: At the start, when Arthur returns the book to Carrie then asks if she'd like to jot down his musings, we can see through the back door's window, and the backyard set's foliage pattern wallpaper is falling off the backdrop. It's visible later as well. (00:02:45)
Continuity mistake: At the start when Arthur is on about Al Gore, he has chicken in his hands, but when the camera cuts, he now has a fork and napkin. (00:01:35)
Continuity mistake: At the beginning Carrie is dishing out potatoes from a bowl with a large black spoon, and as the camera changes angles her arm has not moved and she is now holding a small fork getting chicken from a different bowl. (00:01:40)
Continuity mistake: At the beginning during dinner, Carrie hands Doug the salad bowl, then puts the spoon into the bowl and hands it to him again in the next shot. (00:02:25)
Continuity mistake: At the start Doug puts a drink down after taking the top off. Camera cuts and the lid is back on. (00:03:30)
Continuity mistake: Doug sits down near the start after getting a drink and proceeds to pour a drink. When it cuts he has instantly stopped pouring the drink. (00:03:40)
Continuity mistake: At the dinner table while talking about being obscure at work, Doug starts pouring his beer into a glass. But immediately in the next shot he is finished pouring and both bottle and glass are on the table. A moment later he picks up the beer and pours it into the glass a second time. (00:03:45)
Continuity mistake: At the start of the episode when Arthur is on about kissing ass, the way his napkin is positioned constantly changes. (00:03:45)
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)
Visible crew/equipment: When Arthur is saying about how someone should be able to go out their house and turn left or right, he looks up and the top of the set and the set lights are visible in his glasses. (00:02:45)
Continuity mistake: When Doug and Carrie are in the bedroom discussing the health benefits of Doug losing weight, Carrie's arm is constantly switching from in her lap to on her head. (00:06:50)
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)
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)
Continuity mistake: When Carrie asks Arthur how a legless dummy will protect her on the subway, her arms are either on the table or off, depending on the shot. (00:10:30)
Continuity mistake: When Doug first enters the gym, you can see a price tag hanging off of the front of his weight belt. In the shots following this, the price tag is gone. (00:11:10)
Continuity mistake: When we first see Doug at the gym, the sweat stains on his shirt keep changing size. (00:11:30)
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)
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.