Revealing mistake: When Doug stops to tie his shoe outside the car dealership and Carrie walks up to grab him, you can see the fake back-drop sway with her motion. (00:08:35)
Continuity mistake: When making a PB&J sandwich, the jar of peanut butter rotates by itself. Note the label. (00:09:30)
Revealing mistake: When driving the Mini, the reflection on the bonnet never changes despite being on the highway. (00:12:00)
Continuity mistake: When driving the Mini, the cars around them are constantly changing. (00:12:05)
Visible crew/equipment: After the first drive in the Mini, a stage light is reflected clear as day in the bonnet. (00:13:20)
Visible crew/equipment: When Richie is explaining about getting rid of the car, tape markers are visible by Doug's feet. (00:14:40)
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.