Continuity mistake: The bag Doug puts on Arthur's hospital table is a lot bigger than the bag he got at Gabe's. We know it's not a different bag as he ran from Gabe's, and the bag says Gabe's on it. (00:19:00)
Continuity mistake: Doug closes his magazine at the start. His thumb is in the pages. Camera cuts and now his thumb is on the cover. (00:01:00)
Visible crew/equipment: Carrie begs Arthur to move upstairs, when she does, a tape marker is visible on the floor by the stairs to the basement. (00:05:30)
Continuity mistake: When arguing with Arthur about moving upstairs, Carrie has her hand on a chair or the table depending on the shot. (00:05:50)
Visible crew/equipment: When Carrie walks into the bedroom at the start, 2 tape markers are visible on the green carpet. (00:06:55)
Visible crew/equipment: When Doug says he doesn't want to look after Arthur in the bedroom, there are 4 tape markers visible on the floor at the foot of the bed. (00:14:30)
Continuity mistake: When Carrie is on the phone to change the insurance cover, her hair is either tucked behind the phone or covering the phone depending on the shot. (00:15:50)
Continuity mistake: When arguing about who is going to die first, the amount of whipped cream around Doug's mouth changes instantly. (00:19:20)
Continuity mistake: When Doug is speaking to Spence and Deacon about Carrie throwing up, Spence takes the cap off his beer which reappears a few shots later. (00:02:45)
Visible crew/equipment: In the church just as the minister tells Doug and Carrie to take a seat, two white tape markers are visible on the red carpet. (00:05:30)
Continuity mistake: Carrie tells the preacher that she and Doug met at a bar. When she hears the lie about her falling on Doug as a kid, she believes it and acts as if that was their only meeting before the bar. But in the episode "Road Rayge", she mentioned a junior high school dance where Doug asked her to dance.
Revealing mistake: When Carrie is talking about going to the fertility clinic, you can see the fridge behind Doug through the door. If you look closely just before the camera changes a boom shadow moves across the fridge. (00:01:35)
Continuity mistake: Doug gets into a mess with the sellotape wrapping Carrie's Christmas present. However when she walks downstairs it's on the table, absolutely fine. (00:02:20)
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.