Visible crew/equipment: When the squad's siren gets stuck blaring, a crowd gathers around the squad with Andy, Barney and Rogers, and when the crowd disperses we can see at least three chalk T-marks on the ground.
Visible crew/equipment: At the start of the last scene, when Andy and Rogers walk out of the courthouse the reflection of a crew member is visible on the surface of the car, and as Rogers drives away we can see the reflection of the camera dolly on the car, and also the shadow of the boom mic on the ground.
Visible crew/equipment: When Rogers shows Andy the statistical chart and they get a call about the domestic violence, the two chalk marks can are visible on the floor below the bulletin board to its left, which is precisely where both Rogers and Barney have stood at different times during this scene.
Other mistake: Notice the license plate of Rogers' car is "DC 269," which is the same plate number as Andy's squad in a few season 1 episodes such as "Ellie Comes to Town" and "A Feud is a Feud."
Continuity mistake: On the curb outside the courthouse there's a fire hydrant beside the pole, which not only changes its position on the sidewalk in different scenes, but it has also vanished at the end when Rogers drives away, as Barney crosses the street.
Continuity mistake: While the squad's siren switch is stuck in the on position, Barney and Rogers get out of the car to open the hood, and suddenly there's a pencil sticking out of Barney's shirt pocket for Rogers to use to tap the condenser box, to stop the blaring siren.
Visible crew/equipment: When Andy and Barney are in the squad car talking about the crooked stop sign, in both of their closeups we can see the black tape that had been applied to the chrome trim of the squad's windows, presumably to prevent reflections. That tape is gone from the driver's side when they park in front of the courthouse.
Revealing mistake: The woman in the foreground, as the siren is stuck, is carrying a paper grocery sack from "Hollywood Ranch Market" which was located on Vine St and Fountain Ave in Hollywood, Los Angeles. (00:09:35)
Continuity mistake: At the beginning of the show, Andy notices a wanted poster on the office bulletin board and tells Barney that he had learned from news accounts that the wanted person had been captured. Barney removes the poster and places it in a file cabinet. The next day when Roger reports for duty, he and Barney are looking at the bulletin board and the wanted poster is back up.
Answer: Those "boxes" are valences that used to be very common, before air conditioning. They allow for windows to be open during rain storms. They permit air circulation, without letting the rain in.