
Aunt Bee's Invisible Beau - S5-E27
Revealing mistake: When Clara and Aunt Bee are talking in the kitchen, in the closeups facing Aunt Bee we see the counter, the cabinet, and the wall, but what we don't see are the curtains, the window, or the windowsill, which all should be on that wall. The fake window wasn't placed on the set wall for Bee's closeup.
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.