Other mistake: The end credits list actor Burt Mustin as playing a character named "Jud", but after Howard's introduction to everyone in The Lodge, when Andy and Howard are saying goodbye to all the members, Andy says, "See ya, Burt" to actor Burt Mustin. Then at the start of the voting scene even the subtitles read "Burt" instead of "Jud." (00:09:15 - 00:12:55)
The Barbershop Quartet - S7-E3
Other mistake: The end credits list actor Burt Mustin as playing a character named "Jud", but while Andy and the rest are waiting for Howard to show up for practice, Andy, Wally, and Aunt Bee all use Burt Mustin's real name "Burt" instead of his character's name, "Jud." Then during the rest of the episode Andy still calls him "Burt" a few more times. (00:02:00)
The Barbershop Quartet - S7-E3
Other mistake: At the end of the show Andy talks about the upcoming bowling league. Howard shows Andy his injured left thumb. This should not affect his bowling since Howard is right handed.
Other mistake: Helen is sitting on the first base side, there is a play at the plate and Opie is called out at the plate. Near the end of the show, Helen has a picture she took of the play and it show Opie to be safe. The picture angle is taken almost directly from home plate and at a low angle. The backstop is also blocked half the way up. She would have had to be on the field behind home plate for such a picture. It's also very close up but the camera she is using doesn't have any zoom feature.
Other mistake: Twice during the episode there's a brick wall adjacent to the grocery store which should not be there. First when Clara walks down the street toward Floyd, then later when Andy and Opie also walk down the street toward Floyd. (Shot in the studio not backlot). (00:02:05 - 00:16:55)
A Visit to Barney Fife - S7-E18
Other mistake: After Andy and Barney hear on the radio that another supermarket has been robbed they head over there, and in the shots at the supermarket note that many of the products on the shelves have their brand names covered with tape or scribbled over. For example, boxes of "Mrs. Cubbison's" stuffing/dressing and "Sunshine Cheez-It" are covered with tape. (00:16:40)
Andy's Old Girlfriend - S7-E20
Other mistake: At the cabin, when Andy and Alice are fixing breakfast at the kitchen counter, note the cereal box's brand name is covered (presumably painted), which is still recognizable as "Post Toasties." (Andy Griffith even did commercials for this specific cereal). (00:20:00)
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)
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.
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.