Goodbye Sheriff Taylor - S5-E10
Continuity mistake: While Barney's trying to instruct the new deputy candidates, when he throws a tomato to Otis it splatters on his shirt, and a few things keep changing depending on the angle of the shot - the amount of tomato goo increases/changes position; the suspender's sliding adjuster changes height; the deputy badge changes 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.