Visible crew/equipment: When Carrie come out into the back garden to speak to Doug and Deacon, a boom mic drops into shot very briefly. (00:08:55)
Visible crew/equipment: When Doug tells Carrie "eyes off" someone can be seen moving on the stairs behind them due to a foot appearing in shot and shadows moving. However no-one else is at home other than Doug and Carrie. (00:10:40)
Visible crew/equipment: When Deacon shows up with no shoes he steps inside the house and there is a white tape marker visible where he stops. (00:15:20)
Visible crew/equipment: When we first see Doug on a bike in an overhead shot the camera rig is reflected in a car windscreen. (00:17:50)
Visible crew/equipment: When Carrie gives Doug the card, the top of the set is visible behind them. (00:10:40)
Visible crew/equipment: When Doug says about Kelly being in the wrong he puts his arm up in the air which shows the top of the set and some equipment. (00:14:00)
Visible crew/equipment: When Doug arrives at the airport and we see the two gentlemen writing names on a board, everyone has tape markers at their feet. (00:03:20)
Visible crew/equipment: When Carrie comes back saying she can't swap the flights, there are various tape markers visible on the floor. (00:19:00)
Visible crew/equipment: When Doug gets the beach ball, all the stage lights are reflected in it. (00:14:00)
Visible crew/equipment: When Doug and Arthur arrive in the City to talk to Carrie, yellow tape markers are visible on the floor by Doug's feet. (00:15:10)
Visible crew/equipment: We see Doug slide past the camera at Carrie's hospital appointment. When he does, a boom mic is reflected in the picture behind them. (00:11:50)
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.