Ernest T. Bass Joins the Army - S4-E3
Revealing mistake: When Barney picks Andy up and they get back to the courthouse, they find Ernest T out of the jail cell again and breaking more windows, and in the shot of Ernest T leaping in the air we see it's actually a flipped shot - note the backwards "snack bar" in the window (that store is later when Ernest T tries to break the new glass for the courthouse).
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.