The Statue - S7-E23
Other mistake: At the end of the ceremony, when Aunt Bee finally unveils the life-size statue of Seth Taylor the prop jiggles (presumably it's plastic) back and forth on the base, even though we know it's carved out of a block of granite (stated by Howard earlier in the episode) by a stonemason. (00:05:00 - 00:22:55)
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)
The Statue - S7-E23
Continuity mistake: At the end, when Aunt Bee concludes the reading of the minutes and mentions the improvement fund, she's holding the blue folder up in her hands with its notepapers inside, but in the next shot one of the notepapers is suddenly flipped over the top of the folder. (00:23:45)
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.