High Noon in Mayberry - S3-E17
Deliberate mistake: While Luke and Andy are in the house, Barney, Gomer, and Otis are hiding on the porch, and Barney has the idea to turn off the lights in the house, so he goes to the circuit box located beside the windows on the front porch and switches the lights off. The problem is that this circuit box is located on the porch only for this episode as a plot point.
Continuity mistake: While Peggy's waiting in front of the drugstore she's splashed by a car driving through a puddle, and when she shows up at Andy's house the stains on her dress are different.
Visible crew/equipment: While the fake band members begin to practice before the mayor shows up, in Andy's closeup right before he tries to stop Phil's flamboyant playing, the reflection of filming equipment is visible on the surface of Andy's tuba.
Visible crew/equipment: At the start, when Andy and the rest of the band are at Floyd's, we can see the black tape mark on the floor behind the barber's chair where Floyd's standing. Watch Floyd walk away from the chair for a moment, and when he goes back he looks down twice to make sure he's standing at his mark.
Other mistake: When Andy, Barney, and Opie switch Aunt Bee's kerosene tasting pickles with the store bought pickles during 'operation pickle switch', they put the store pickles in Aunt Bee's mason jars with silver lids and Aunt Bee's pickles in the store's jars with labels and white lids. Their plan is to give away the store jars with Bee's pickles inside, and we see Barney placing the last of the store jars into his bag, but when Barney stops the driver to give him his safe driving award, we see Barney giving away the mason jars with silver lids even though it's the store jars he should be getting rid of.
Visible crew/equipment: When Wilbur's in his car he thanks Andy and the rest of the men for all the shoe sales. As he drives off, the reflection of filming equipment is visible on the surface of the car, and when the car is offscreen multiple chalk marks are visible on the sidewalk, at the feet of the actors.
Visible crew/equipment: When the men have the misconception that Wilbur is a TV show talent scout, while they're at the barber shop talking about it, just as Floyd moves his legs to get up from the barber chair we can see the white T-mark at Jason's feet.
Revealing mistake: During The Manhunt in the woods, Andy, Barney, the trees, etc., cast long shadows due to the overhead sun, but the problem is that we also see Andy's and Barney's shadows being cast in another direction as well, due to the set lighting on location.
Deliberate mistake: During scenes at Walker's Drugstore there are many magazines by the wall near the door, which have tape/paint hiding part or all of their cover titles. Examples are: all the Sports Illustrated have tape across the word "Illustrated", the House and Garden has its "House &" covered, and Time magazine's entire title is blacked out.
Mayberry Goes Hollywood - S1-E13
Visible crew/equipment: Before the producer's arrival in his car, when Barney's telling Andy that it'll serve him right if they don't put him in the picture because Andy's not wearing his new uniform, we can see the moving shadow of the boom's hinge and mic on Andy's shirt.
Continuity mistake: When Barney and Andy are looking at Barney's picture on the front page of the newspaper, first, the article has nothing to do with Barney or the arrest, and second, when Barney takes the paper from Andy he folds it with the front page on the outside, and the page we see is not the same front page from the closeup.
Continuity mistake: When Andy locks himself in the jail cell, the key is lying flat on the short wall, but after Barney locks himself in the other cell, the key has turned itself upright on its side.
Other mistake: After Barney gets annoyed over Andy's lack of confidence in his ability to be a sheriff, Barney gets on the phone with Sarah and asks to be connected to the mayor of Greendale, then it cuts to an exterior shot and the parked car has plate number "DC 269," the same as other vehicles in some earlier episodes.
The Clubmen - S2-E10
Other mistake: After Andy turns down the offer to join the club the two men leave, and when it cuts to them driving away we see the license plate number is "DC 269," the same as some other vehicles in previous episodes.
Visible crew/equipment: At the end, when Andy opens the door to Floyd's so Ed Sawyer can get a haircut, the set's large spotlight is reflected on the door's window.
Visible crew/equipment: While Floyd's napping in the barber chair, Andy's reading the newspaper, and when Barney walks in the actor's two curved chalk marks are visible on the floor, where he will stand.
Visible crew/equipment: While Barney's in the barber chair, Floyd starts talking about how he'd tried living at the back of the barber shop, and when the shot faces Barney we can see the reflection of a crew member's legs moving around - the only ones in the shop are Barney, Floyd, and the two men seated by the corner.
Continuity mistake: When Andy and Mrs. Silby are walking into the courthouse, in the exterior shot of Andy opening the door he's wearing the baseball mitt on his left hand, but when it cuts to the interior shot Andy's opening the door with his left hand and the mitt has vanished.
Aunt Bee's Brief Encounter - S2-E9
Other mistake: Mr. Wheeler's license plate number is 'DC 269' which has been used in earlier episodes. Then at the end, when the locks-keys man pulls his truck into the driveway, it has plate number 'VT 772' another prop plate seen in "Barney on the Rebound" when the pretty woman drives into town.
Visible crew/equipment: While Andy's in Opie's bedroom when Opie's thinking about running away, the moving shadow of the overhead boom mic is visible, first on the wall behind Opie's headboard and then on the door. Also note, when Andy shows Aunt Bee how to hold the baseball bat and moves her into position, we can see a long white tape mark on the grass (since it's possible that Andy placed the tape, however unlikely, it's being noted here).
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.