Visible crew/equipment: At the end in the theatre, a boom mic drops into shot. (00:19:00)
Head First - S1-E6
Revealing mistake: When Doug is making breakfast it's clear there is no flame coming from the gas hob. (00:08:15)
Character mistake: Doug claims to be a Pisces, however during this episode when entering his birthday into the alarm system to disarm it he calls out 2-9-69 which shows he would be an Aquarius.
Continuity mistake: While Arthur tries to convince Spence about the card, when Spence bites into the fake cookie he's holding it in his left hand and holding the plate in his right hand, but in the next shot facing Arthur the items have switched hands. (00:16:30)
Continuity mistake: When Carrie takes the first lot of Hofferman's pictures off Doug, she's holding them portrait. Camera cuts and she's holding them landscape. (00:05:30)
Continuity mistake: When Doug and Carrie are told that mold isn't covered by their insurance, Doug has his fist clenched by his mouth. Camera cuts and now it's open palmed and on the side of his head. (00:01:00)
Continuity mistake: When discussing the Superbowl at IPS, Danny has a roll in his hands. Camera changes and he picks it up again. (00:04:30)
Revealing mistake: When Carrie knocks the cards over at the start, some stay in the pyramid shape despite being knocked over, revealing the cards are glued together. (00:03:15)
Audio problem: When Doug goes to the door to call for Uncle Hank, the audio is over dubbed. (00:04:05)
Continuity mistake: When Carrie is talking to Tina about playing 3 holes, her hands are outstretched. Camera cuts and her gloved hand is now on her hip. (00:16:40)
Continuity mistake: Arthur gives Holly a book at the start and she has it in her right hand. Camera cuts and now it is in her left hand. (00:03:20)
No Orleans - S4-E16
Continuity mistake: When Spence is seen eating a snow cone for the first time, the level of it changes between shots. (00:06:10)
Continuity mistake: Arthur grabs a spoon to have some of the brie, camera cuts and he grabs the spoon again. You hear him take the spoon twice as well as it clangs on something. (00:04:15)
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 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)
Audio problem: When the nurse compliments Doug on his costume, she taps him on the arm twice. However the sound is out of sync. (00:08:00)
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.