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: 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)
Revealing mistake: While Goober is up in the tree with the fake owl, when Andy and the Darlings leave the house they are standing on the porch, and after seeing the 'second omen' the Darlings leave the porch and head to their pickup truck, but what's funny is that we can see their reflections in the window behind Andy as they all walk off camera, turn around and just stand there, and we can still see those actors' reflections in the shots facing Andy and Aunt Bee even as the Darlings supposedly drive off.
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.
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: 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)
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)
A Visit to Barney Fife - S7-E18
Continuity mistake: While Andy and Barney are sitting on the porch, Barney's clip-on striped bow tie is clipped on the opposite way (stripes flip) when Ma and her family get back from "shopping." (00:14:45)
Visible crew/equipment: At the start, just as Mr. Hampton walks into the office and pushes the door closed, we can see a crew member's hand reaching through the doorway to help close the door. (00:00:55)
Visible crew/equipment: After the squad car is hit from behind Andy rushes over to Goober, and a T-mark (it's a bit darker than the color of the dirt road) becomes visible on the ground (just as the woman gets out of her car) at the bottom of the screen, right behind Andy's feet. (00:01:25)
Aunt Bee's Crowning Glory - S7-E5
Visible crew/equipment: When Andy removes the mirror from the wall (for Aunt Bee), the reflection of a studio light is visible in the mirror just as Andy starts moving it. (00:18:40)
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)
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)
Visible crew/equipment: When Aunt Bee and Opie leave the house and get in the car, the reflection of crew and filming equipment can be seen on the surface of the car.
Visible crew/equipment: Just as Aunt Bee and Opie pull up in front of Wally's station, the reflections of crew and equipment are visible on the surface of the car. (00:06:00)
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.
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)
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: 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.