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)
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)
Maybe Baby - S1-E25
Continuity mistake: When Doug tells Deacon that he and Carrie are trying for a baby, Deacon has both hands on his knees, camera changes and he's got 1 hand and 1 elbow on his knees. (00:09:50)
Continuity mistake: When Danny storms out he is not through the door - the camera changes and the door is closed and Danny is gone. (00:18:20)
Continuity mistake: When Arthur is listing the genres of the film, his legs go from on the floor to one raised and back between shots. (00:02:10)
Fixer Upper - S1-E12
Continuity mistake: At the start, the items in Carrie's hands swap hands instantly. (00:00:10)
Continuity mistake: When the guy chucks his cheque book into the basket, it reappears in his hand a second later. (00:11:20)
Road Rayge - S1-E9
Continuity mistake: When driving Doug home, look out the back window to see 2 red car lights appear instantly and out of nowhere. (00:12:20)
Road Rayge - S1-E9
Continuity mistake: When trying to get Doug to drive after the Jets' game, Ray's press pass goes from white to green instantly. (00:09:20)
Continuity mistake: In the first classroom scene there are two people sat to Doug's right. These people then vanish a few seconds later. (00:07:55)
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.