Continuity mistake: At the beginning of the show, Barney and Andy are sitting inside a 1964 Ford patrol car. When they are shown up close inside, you can tell it is not the likeness of the same car. The back glass windows look more like a 1960 model possibly. It is definitely not a 1964.
Other mistake: Barney's cousin is coming to Mayberry but gets off the bus in a previous town and misses it. Barney and Andy go out to find him, leaving from Mayberry. They are driving down the road when they come upon his cousin. The cousin is walking in the same direction as Andy and Barney are going, which would have meant he was walking away from Mayberry when he should have been walking towards it.
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)
Emmett's Brother-in-Law - S8-E18
Visible crew/equipment: When Ben accompanies Emmett on his first call to Andy, just as they both enter the courthouse, we can see a crew member's hand outside reach out to catch the door as it closes. (00:12:40)
Factual error: The badges they wear don't match the badges on their patches. If there's a badge on the patch, it's supposed to match their actual badge.
Andy and the Gentleman Crook - S1-E21
Character mistake: When Barney and Gentleman Dan play gin, Barney deals only 8 cards. In gin, the proper amount would be 10.
Revealing mistake: Before Barney jumps out from behind the crate to surprise the two cops in the alley you can hear the director yell out "Don".
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: Opie accidentally drops and breaks the bottle of perfume and doesn't realise it's colored water, a bottle of perfume would smell for days.
Other mistake: In some episodes Barney can sing beautifully, but in others he can't sing at all.
Andy and Opie, Housekeepers - S1-E23
Continuity mistake: Andy is sitting on the couch, next to him on the end table is a glass and what appears to be a dark bottle like a beer bottle. They cut away, when they return to Andy on the couch the glass and the dark bottle have disappeared.
Aunt Bee Learns to Drive - S6-E21
Other mistake: Aunt Bee says that Andy had to drive her to Clara's and Andy says it's not a bother, but Clara has always lived right next door.
Opie's Group - S8-E9
Visible crew/equipment: While Opie and his band are playing at the party there are shots of the kids dancing, and in the shot of a girl with blonde hair wearing a blue dress, the camera angle is low and we can see the top of the set. (00:10:25)
Visible crew/equipment: After the shot of Milly holding hands with Howard (holding a picnic basket), when it cuts to Goober sitting in front of the courthouse the camera angle is low, so the wood framework and lighting equipment at the top of the courthouse set is visible, just as Andy walks over to Goober. (00:05:30)
Other mistake: During the scenes at Elmo's shop, where Aunt Bee's pictures are developed, it's obvious that this is actually Emmett's Fix-It Shop. We can see the window of the shop next door through Elmo's doorway, and recognize it as the store next door to Emmett's shop which has "Furniture, New and Used" on its front window.
Visible crew/equipment: After Andy hatches the plan to have Aunt Bee and her friends pick up their photos at the same time, in the following shot just as Andy walks into the shop, the shadow of the boom mic is moving beside the doorway, at the top left side of the screen.
Visible crew/equipment: After Aunt Bee's flight lands, when it cuts to the hotel lobby there's rug on the floor extending from the front desk to the stairs, but in the next shot as Aunt Bee and her friends walk into the lobby that rug is gone (this doesn't refer to the small rug behind them). Also note the tape mark on the floor where the bellhop stops the luggage trolley.
Visible crew/equipment: When everyone is at the airport before the flight some chalk T-marks are visible on the floor, such as the one between Andy and Helen when Andy takes the picture of Aunt Bee and her friends. (00:08:45)
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.