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: 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: The Saskeys are Doug's neighbors in this episode despite them having moved away back in the season 2 episode "Soft Touch" (00:02:30)
Continuity mistake: When we first see Doug at the gym, the sweat stains on his shirt keep changing size. (00:11:30)
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 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)
Strike Out - S3-E7
Continuity mistake: When Arthur sits down for his meal, there is a bottle on the table. The lid is either on or off depending on the shot. (00:03:30)
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)
Strike Out - S3-E7
Continuity mistake: After Doug says "Let's hit the streets", Doug, Arthur, Deacon and Kirby go on a mini crime-spree. During this montage each characters outfit completely changes several times. They are also dressed very lightly for late October/early November in New York. (00:13:00 - 00:14:00)
Strike One - S3-E5
Continuity mistake: When Doug has a look at the bumper sticker Spence has, there is one fold in it, but when it cuts there is now three. (00:14:40)
Continuity mistake: When Doug says that there is nothing going on at work, his hands are flat on the table, camera cuts and he's got one up in the air. (00:06:50)
Continuity mistake: When the beach ball sails over the fence, Doug picks it up and holds it to his face, making his right side of his shirt wet. Throughout that scene, his shirt is dry, and then wet again.
Continuity mistake: When Carrie turns round and tells Doug that she's telling Kelly about the card, the bottle top on the table is constantly changing position without being touched. (00:11:40)
Continuity mistake: This is a very tricky one to spot as it happens as the shot is fading into black to a scene change but it is noticeable without the use of pause and/or slow motion. When Doug and Deacon are outside talking, Carrie comes out and then proceeds to walk back in and and audibly shut the back door. However as Doug and Deacon walk back in as the shot is fading away, the door is wide open. (00:09:30)
Continuity mistake: When Doug and Deacon both volunteer for the calendar the magazine on the bench is constantly changing position without being touched. (00:05:10)
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)
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.