Visible crew/equipment: Lighting equipment is visible in the lower left corner of the shot of the outside of the Heffernan home. (00:14:42)
Continuity mistake: When Arthur tells Carrie that he has jury duty, Carrie places her used coffee mug on the counter beside the mug tree, then her used mug vanishes, the mugs hanging on the tree change, and the coffeemaker moves. (00:04:30)
Continuity mistake: The lady walks away from Carrie's desk with some flowers. When she does, there are three petals on Carrie's desk which disappear without being touched. (00:06:10)
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)
Continuity mistake: The baseballs on the stand move apart when Doug picks up the first ball to win Carrie a frog at the boardwalk. (00:17:24)
Continuity mistake: The guy on the stall puts 3 balls down in front of Doug in a line. Doug picks up the ball furthest from the guy. Camera cuts and now Doug has the ball closest to the guy. (00:17:30)
Continuity mistake: The first time Doug throws a ball at the bottles, we see one bottle remain lay down on the stand. Camera cuts and this bottle has disappeared. (00:18:00)
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)
Visible crew/equipment: When Doug uses the ridiculous RV horn and startles Carrie awake, as she walks from the sofa to the front door, the overhead studio lights are visible at the top of the screen. (00:11:40)
Deliberate mistake: When Arthur enters the living room playing the harmonica, it is obvious he is not actually playing it. (01:23:10)
Character mistake: In this season we see Doug smoking cigars and cigarettes. However in S2 E1 Queasy Rider, Doug is shown to absolutely detest the smell and taste of smoke from these. It is highly unlikely that he would then take up smoking.
Revealing mistake: At the start, when Arthur returns the book to Carrie then asks if she'd like to jot down his musings, we can see through the back door's window, and the backyard set's foliage pattern wallpaper is falling off the backdrop. It's visible later as well. (00:02:45)
Continuity mistake: At the start when Arthur is on about Al Gore, he has chicken in his hands, but when the camera cuts, he now has a fork and napkin. (00:01:35)
Continuity mistake: At the beginning Carrie is dishing out potatoes from a bowl with a large black spoon, and as the camera changes angles her arm has not moved and she is now holding a small fork getting chicken from a different bowl. (00:01:40)
Continuity mistake: At the beginning during dinner, Carrie hands Doug the salad bowl, then puts the spoon into the bowl and hands it to him again in the next shot. (00:02:25)
Continuity mistake: At the start Doug puts a drink down after taking the top off. Camera cuts and the lid is back on. (00:03:30)
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.