Revealing mistake: 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)
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.