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)
Factual error: We know from 2 episodes prior (Man in a Hurry) and generally mentioned throughout the show, the only filling station in Mayberry is Wally's Filling Station. In this episode Mayor Stoner is accused of only putting the new underpass in Mayberry because it will be next to Mayor Stoner's brother's filling station.
Continuity mistake: The fuel pumps at Wally's gas station (regular and super) switch sides between numerous episodes.
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.