Frozen Pop - S2-E15
Continuity mistake: When Arthur comes in and shuts Doug's window, Doug's hand goes from open to closed between shots. (00:06:00)
Frozen Pop - S2-E15
Continuity mistake: When Carrie is talking to Arthur at the start, Arthur's hands are in his lap in one shot and clasped together in the next. (00:02:45)
Continuity mistake: Deacon states on a few occasions in this episode that him and Doug started at around the same time at IPS. However, in later episodes it's revealed Deacon is senior to Doug by quite a bit in flashbacks to how Doug got his job. (00:10:00)
Continuity mistake: When talking to Doug in the back yard, Carries hands are changing position between instant cuts. In some shots they're together in her lap in others she's holding Doug. (00:04:00)
Court Date - S1-E17
Continuity mistake: When Doug, Carrie and Ritchie are talking about the cop having not called her yet, Doug's hands are either on or off the bottle of beer, and the level of beer/foam changes between shots too. (00:16:30)
Court Date - S1-E17
Continuity mistake: When Arthur storms out saying that was his drug speech, Carrie has her wrists in the table in one shot and under the table in the next. (00:15:10)
Hungry Man - S1-E21
Continuity mistake: At the very end when Carrie walks out the level of foam on Doug's beer bottle changes between shots. (00:21:20)
Continuity mistake: When Arthur tells Carrie that he has jury duty, Carrie places her used coffee mug on the counter beside the mug tree, then her used mug vanishes, the mugs hanging on the tree change, and the coffeemaker moves. (00:04:30)
Continuity mistake: While Doug is sitting on the corner armchair reading, Carrie's on the sofa holding her coffee mug and magazine, but when she tells Doug to join her on the sofa her coffee mug vanishes from her hand, and then it reappears. (00:01:00)
Continuity mistake: In the first shot inside Arthur's house, Carrie and Doug are standing together when Carrie mentions how tough it must be for Arthur. It then cuts to Arthur sitting at the table, with Doug right there sitting beside Arthur. Next shot Doug is still standing with Carrie as she urges Doug to spend some time with Arthur, so this is when Doug walks over to his father-in-law. (00:07:00)
Train Wreck - S1-E20
Continuity mistake: When Doug sees Arthur's headstone in the yard, the dark opaque flap of the dog door is missing from its frame, but it reappears as Doug enters the kitchen. (00:06:40)
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.
Head First - S1-E6
Continuity mistake: When Doug and Deacon are having breakfast in the kitchen, the door keeps changing from open to closed with no-one touching it. (00:08:30)
Continuity mistake: When Doug Tells Carrie to pack for Atlantic City the cushion is either on or off her lap. (00:17:15)
Deconstructing Carrie - S7-E15
Continuity mistake: While Carrie was being harassed by the construction workers, Doug was watching by looking out the door to the right. In the next shot, you see Carrie through the window, coming in from the left.
Continuity mistake: The garage door in the outside view has 4 windows. Switch to the inside scene and it has 3.
Food Fight - S4-E13
Continuity mistake: When Spence and Becky arrive near the end, Doug's coat on the floor changes position with no-one touching it. (00:17:20)
Food Fight - S4-E13
Continuity mistake: When Doug is checking the mail while telling Carrie about checking the food twice a week, the letter at the back of the pile is red. Camera changes and now it's a normal envelope. (00:05:30)
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.