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.
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.
Answer: It was a joke. They were supposed to be dumb hillbillies.
Both. According to later interviews with both Andy and the Dillards band (the real name of the band that played the parts of the Darling children, of which Charlene (Maggie Peterson - Mancuso was also a member) the reason the Darling boys never had any lines was because they would've had to have been paid if they spoke. In return for appearing without lines or pay, Andy promised to get as much of their music onto the show as possible which, he did. This obviously gave them a lot of exposure.