Fixer Upper - S1-E12
Continuity mistake: In "Richie's Song" Doug asks Spence if the M&Ms Spence found under the seat in the car are regular or peanut, Spence answers peanut, and Spence then eats them. In this episode, Spence somehow isn't allergic to peanuts. (00:08:10)
Continuity mistake: In many episodes, scenes are shot on the front porch. It is often shown as a large porch that spans a great deal of the front of the house. But shots of the house exterior from a distance show a small set of stairs with very little space near the door. This isn't a porch on the back of the house as it has been shown to only be door to a small yard.
Queasy Rider - S2-E1
Visible crew/equipment: When Doug comes into the garage and falls over for the first time, a tape marker is visible on the floor. (00:07:05)
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)
Window Pain - S5-E2
Visible crew/equipment: When Doug gets into their Jeep after meeting the new neighbours, a man and the backstage can be seen in the passenger side mirror. When the shot cuts back to Doug again it's darker but when he backs up, the reverse lights of the Jeep reveal the man and stage again. (00:26:55)
Queasy Rider - S2-E1
Continuity mistake: When Doug tells Carrie that he is getting rid of the bike, Carrie has got a fag in her hand and her hand is up by her head. Camera cuts and now her arm is down across her lap. (00:15:40)
Continuity mistake: In the scene where Doug's father is shutting off the alarm, he does not enter 2965. We know this is the proper code, because it is Doug's birthday, and it worked when he entered it. It looks like he maybe enters 1459.
Continuity mistake: In this episode, when Doug and Carrie are taking an adult literature class together, Doug comes down the stairs with his book in his hand, but when he sits on the couch he only has his book pillow and reading lamp, and runs back upstairs to get his book.
Continuity mistake: When Carrie takes Doug into the back yard to yell at him, after she is done swinging at him, she puts her hand up to her chin. First it's her left, then right. It changes between shots. Each time the camera goes back to her it happens.
Visible crew/equipment: When Doug is getting a lift from the Priority Plus driver after being rescued from being stuck in the drive-thru, you can see the camera truck (the truck they use to film exterior road/driving shots) in the reflections of the glass windows of buildings behind Doug when the camera cuts to him.
Revealing mistake: In the final scene of the episode, Arthur is searching for used bottles of nasal spray. He empties the contents of the trash can on the kitchen floor and finds one after sifting through the trash. However, the bottle had already been positioned on the floor next to the trash can before he picked up the trash can so it would be easy to find when the trash was emptied on the floor.
Continuity mistake: Just as Doug persuades Deacon to go with him to Carrie's birthday dinner, his position changes completely. (00:10:20)
Continuity mistake: In the flashbacks in this episode, Carrie and Doug meet for the first time in a bar, but in the episode "Road Rayge", Carrie mentions a junior high school dance where Doug asked her to dance.
Continuity mistake: When Sara is in the shower you can see in her shadow, creases from her swimsuit. (00:16:30)
Revealing mistake: When Doug and Deacon are driving round trying to find Gabe's Pizzeria, they get into the car and pull out into the road. However if you look at the traffic, they would have hit at least 2 cars in doing so. But they don't, showing it's a projection. (00:14:40)
Deconstructing Carrie - S7-E15
Revealing mistake: When Doug flies off the bike and across the screen, in the background is a palm tree, which obviously would not grow in New York. (00:14:20)
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.