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.
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: 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.