The Arrest of the Fun Girls - S5-E28
Other mistake: When Andy, Helen, and Thelma Lou are parked in front of the courthouse the lights turn on, and in the closeups from outside both sides of the car, note that both triangular vent windows are missing - the two doors are actually open for these closeups. This happens again later, when Andy and Barney are each apologizing individually to Helen and Thelma Lou.
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.