Aunt Bee's Big Moment - S8-E23
Continuity mistake: When Opie asks his father how to spell the word "Renaissance," Aunt Bee holds a plate of assorted cookies, but when she stands up and heads to the kitchen the type of cookies on her plate have changed.
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.