Visible crew/equipment: After Doug gets Arthur out of the house he takes him to a restaurant, and as they both take their seats, the two sets of double tape marks are visible on the carpet under the table. Then, when Doug stands up another small tape mark is visible at his feet. (00:06:30 - 00:08:00)
Frozen Pop - S2-E15
Visible crew/equipment: When Arthur puts his big red jacket on and goes downstairs, Carrie follows and tape markers are visible on the floor by the door. (00:16:30)
Frozen Pop - S2-E15
Continuity mistake: When Doug has just gone back to bed and Arthur walks in, Doug's position on the bed changes instantly. He swaps sides. (00:05:35)
Frozen Pop - S2-E15
Continuity mistake: When Arthur comes in and shuts Doug's window, Doug's hand goes from open to closed between shots. (00:06:00)
Frozen Pop - S2-E15
Continuity mistake: When Carrie is telling Arthur to move upstairs, the blanket Arthur has got changes position instantly. (00:02:40)
Frozen Pop - S2-E15
Continuity mistake: When Carrie is talking to Arthur at the start, Arthur's hands are in his lap in one shot and clasped together in the next. (00:02:45)
Frozen Pop - S2-E15
Continuity mistake: When Doug goes back into the house at the start he opens the door with his right hand. Camera cuts to inside the house and now it's his left hand on the door handle. (00:01:35)
Party Favor - S2-E13
Continuity mistake: When Doug is mimicking the Japanese chef, the towel on the side moves positions without being touched. (00:01:00)
Party Favor - S2-E13
Continuity mistake: At the start, Doug has a spatula in one hand and a whisk in the other. Camera does an instant cut and they have swapped hands. This is not to be confused with when he is messing about and you see him swap the items between hands. (00:00:50)
Frozen Pop - S2-E15
Visible crew/equipment: While Doug is scraping the windshield in the wide shots, the end of the street backdrop and production equipment are visible at the upper left side of the screen. (00:01:15)
Visible crew/equipment: Doug whisks Carrie to the boardwalk straight from the train after work, and when Doug points to the booth with plush frogs, a camera is visible on the left side of the screen as it moves out of shot. (00:17:20)
Revealing mistake: After Doug successfully trades vacation days with Roger, it cuts to Doug's IPS truck stopping in traffic. The next few shots were filmed on the streets at Sony Pictures Studios lot. In the shot of the IPS truck's side-view mirror (as the RV approaches), note the signs on the building read "Keaton Building" and "Sound Department Courtyard." After the flashback, another building sign reads "Durante Building." These buildings are located at the north side of the studio in Culver City, CA. (00:03:20)
Soft Touch - S2-E22
Visible crew/equipment: At the start, in the shots facing Carrie and Doug while they're in the car, we can see a screen atop the roof on the driver's side of the car with its reflection on the windshield (and its shadow on the car's hood). This is gone in the overhead long shots. (00:00:10)
Continuity mistake: When Doug and Arthur are sat at the dinner table, Arthur's hands go from on the table to clasped between shots. (00:07:45)
Queasy Rider - S2-E1
Visible crew/equipment: When Doug and Carrie are arguing over the bike a white tape marker can be seen on the floor. (00:09:20)
Queasy Rider - S2-E1
Continuity mistake: When Spence is talking to Carrie at the restaurant at the start, Spence has his hands out in front of him. Camera cuts to another angle and now he has his drink in his hand. (00:01:45)
Queasy Rider - S2-E1
Continuity mistake: When Doug is checking Arthur's CV at the start, the items on the counter keep moving - noticeably the mayo and lid. (00:00:25)
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.