Other mistake: In some episodes Barney can sing beautifully, but in others he can't sing at all.
Other mistake: The radio communications consoles on the side desk in the courthouse have a few pieces of black tape covering their brand name (though we know it's Motorola from the visible "M" logo).
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.