The Statue - S7-E23
Audio problem: When Aunt Bee is introduced at the start of the ceremony, in the first wideshot as she begins her speech we hear her say, "Friends, neighbors, and fellow citizens of Mayberry" but it's not in sync with her mouth movements. (00:18:45)
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.