Continuity mistake: At the end, when Andy walks in on Barney showing Gomer how to draw his gun, Gomer draws his gun out of the holster which sends it flying behind him, and in the closeup the bookcase glass shatters, but in the wideshot when they all turn to look the glass is still perfectly intact.
Visible crew/equipment: When the partners of the two men locked up in the jail drive up to the courthouse and park, the reflections of two crew members are visible on the car's vent window.
Continuity mistake: When Gomer drops the gun from the roof it lands on the ground directly in front of Andy and Barney, who are standing on the sidewalk beside the courthouse, but the area where they're standing looks different between the overhead shots and closeups, and their shadows change as well. Additionally, the gun lands with its handle to the left, but in the wideshot the gun is lying the opposite way, with its handle to the right.
Revealing mistake: The jailed criminals take the key and open the jail cell door. The problem is the dead bolt never moves with the key, showing the door was never locked to begin with.
Revealing mistake: Before Barney jumps out from behind the crate to surprise the two cops in the alley you can hear the director yell out "Don".
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.