Aunt Bee's Restaurant - S7-E21
Visible crew/equipment: After Goober orders the chow mein dinner at the restaurant, in the following wideshot of everyone at the table, the curved chalk actor's mark is visible on the floor between Andy's and Helen's chairs. (00:12:20)
Aunt Bee's Restaurant - S7-E21
Continuity mistake: While Jack is trying to help Aunt Bee understand her fear, in a few of the closeups of Aunt Bee and Andy (seated on the couch) note the fruit bowl has moved and the candlesticks have vanished from the server, the large figurine on the left side of the hutch has vanished, and the flowers from the table have also disappeared. Then everything is back in their place. (00:21:10)
Continuity mistake: When Harry hangs the sign on Floyd's shop it reads "For sale Harry Walker real estate," but in the closeup of the sign the word "by" suddenly appears and it reads "For sale by Harry Walker real estate", then it disappears. It looks as if it reappears in the wideshot when Howard stops to read the sign later. (00:01:15)
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)
Revealing mistake: During the scenes with Opie and Dolly in Walt's barn, the horse is wearing a halter with a ring under the adjustable chinstrap, and in every closeup a thin wire (like picture wire) can be seen attached to the ring as it's being pulled from offscreen to have Dolly turn her head away from Opie. Then later when Walt returns and tries to feed Dolly, once again a thin wire attached to Dolly's halter is pulled from offscreen, so she will turn her head away from Walt. (00:05:55 - 00:18:20)
Visible crew/equipment: When Clara and Opie first sit down at the piano to begin the piano lesson, note there's nothing on the floor behind them. Then that shot cuts to the end of the lesson as the camera pulls back for a wideshot, and now there are two tape marks on the floor near the door in the area where Opie and Aunt Bee will be standing shortly. (00:04:15)
Visible crew/equipment: When Flip pays a visit to the Taylors, he and Andy walk over to Opie, and the actors' two tape marks are visible on the floor between Opie and Flip. (00:21:05)
Continuity mistake: When Opie surprises Andy by telling him that he wants to take piano lessons with Clara, he heads to the piano to play what he already knows, and when Opie plays Chopsticks on the piano he's wearing a different shirt.
Visible crew/equipment: After the boys meet Flip Conroy and talk about the football team, when they all disperse and Opie starts walking away the T-mark is visible on the ground.
Continuity mistake: After Howard's appearance on the TV show, when he and Andy get back to Mayberry and talk to Opie on the porch, both Andy's tie and Howard's bow tie have striped designs that are positioned differently once they walk into the house, which could only happen if they were retied. (00:15:00)
Other mistake: Right before Aunt Bee wins the five dollar prize at Wally's station she talks about how she's never won anything, not even the church raffle, and right after she wins she excitedly waves the five dollar bill and says, "I've never won anything before in my life!" However, in season 6 episode "Aunt Bee on TV" the man from the IRS stated that Aunt Bee had won $4,850 worth of prizes on the game show, and even though she sold most of her prizes she kept two of them. So her statement makes no sense even for Aunt Bee.
Other mistake: Twice during the episode the actor who plays Floyd is replaced by his body double. The first time is when Floyd's driving to and away from Wally's station in the earlier scene, and the second is later, when Aunt Bee leaves Floyd's shop and we see Floyd stand up through the window.
Visible crew/equipment: When Aunt Bee drives her car up to the gas pump at Wally's station and then wins the five dollars, the reflections of crew and filming equipment are visible on the surface of her car. Also, note that the chrome trim around the windshield is covered with black tape, presumably to prevent reflection. Additionally, during all the scenes at Wally's station, the back end of one of the quonset huts from the "Gomer Pyle, USMC" set are visible, either at the left side of the screen or in the background.
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.