Continuity mistake: At the beginning of the show, Barney and Andy are sitting inside a 1964 Ford patrol car. When they are shown up close inside, you can tell it is not the likeness of the same car. The back glass windows look more like a 1960 model possibly. It is definitely not a 1964.
Revealing mistake: Barney and Mr Weaver scuffle in the dark. When Andy enters the store he turns on the light, but the light switch is still in the off position.
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.