Other mistake: When Barney, Thelma Lou, Helen, and Andy are in the squad car talking about getting dinner, there's black tape on the chrome trim of the triangular vent window - presumably to prevent the camera/crew's reflections; and the vent window is shut in wide shots, but open in the closeup - also to prevent reflections.
Other mistake: In this episode (aired Sep. '64) Barney tells Andy that 5 years ago on May 16 (1959) he joined the sheriff's office, and later they all have a surprise party for Barney. However, in season 4 "Citizen's Arrest" (aired Dec. '63) Andy tells Barney that he found, "The custody receipt for the first revolver I issued you... August, 1953... 10 years ago last month since I swore you in." Andy even mentions having a party with a cake that has 10 candles on it.
The Education of Ernest T. Bass - S5-E4
Other mistake: After Ernest T. breaks Helen's window Andy shows up, and there's black tape adhered to the chrome trim at the side of the windshield and also the chrome trim beside the vent window.
Other mistake: When the escaped convict takes Barney hostage, they walk away from the lake and Barney drops the piece of paper with the convict's picture. Later, when Andy walks across the grass he finds Barney's cap and the paper lying on the grass, but the problem is that Andy needs to pull up at the center of the paper because it's pinned to ground, presumably so it wouldn't move or blow away for the shot.
Three Wishes for Opie - S5-E14
Other mistake: When Andy asks Floyd and Judd about who he's supposedly marrying it cuts to Helen walking down the street, and the grocery store has only one display window, when it should have two windows, and there's a brick wall adjacent to the window which shouldn't be there. (Shot at the soundstage not backlot).
Three Wishes for Opie - S5-E14
Other mistake: When Barney uses the gypsy lamp and cards in the back room, in the shots with Goober and then with Opie while they're standing in the doorway, the wall opposite them has a barred window, but in earlier episodes that wall has the door that leads outside to the side of the building.
Barney Runs for Sheriff - S5-E21
Other mistake: When Floyd tells Barney that he doesn't want to put Barney's campaign poster in the barbershop window because it spoils the decor, the reflection we see in the mirror behind Floyd is the church's front door and steps. This makes no sense at all because the reflection should be whatever is to the right of Floyd's, but the church is the opposite way - to the left, past the courthouse. The view in the mirror is a painted backdrop, but it's still quite wrong.
Other mistake: When Andy and Helen are in the squad car alone after going to the movies with Gloria, both car doors are closed, but in the closeups from outside both sides of the car, note that both triangular vent windows are missing and we can see the doors' rubber seal - both doors are open for these closeups.
The Case of the Punch in the Nose - S5-E25
Other mistake: In the shots from outside the courthouse, when the front door is opened there's a wall where none should be, presumably to prevent seeing inside the courthouse facade.
Other mistake: After the preacher walks past Andy and Sam, when the two men comment that the preacher's sermons "go on" and are "dry as dust," we can tell that all of the store's glass window panes have been removed, presumably to avoid the crew/camera's reflections as the shot pans across the window to Howie.
Aunt Bee's Invisible Beau - S5-E27
Other mistake: After Andy talks to the butter and egg man in Mt. Pilot, when Andy gets in the squad car with Barney all the car doors are closed, but in the closeups from outside both sides of the car, note that both triangular vent windows are missing - the two doors are actually open for these closeups.
The Arrest of the Fun Girls - S5-E28
Other mistake: When Andy, Helen, and Thelma Lou are parked in front of the courthouse the lights turn on, and in the closeups from outside both sides of the car, note that both triangular vent windows are missing - the two doors are actually open for these closeups. This happens again later, when Andy and Barney are each apologizing individually to Helen and Thelma Lou.
Opie Flunks Arithmetic - S5-E30
Other mistake: When Aunt Bee walks out of the grocery/meat store the door's located between the grocery's one display window and the TV display window, but the grocery store actually has two display windows with its door between the two. This was shot on the soundstage, not backlot.
Opie's Job - S6-E1
Other mistake: In the scenes outside the grocery store with Mr. Doakes, there are a couple dozen cereal boxes in the display windows - Post Grape Nuts, Post Grape Nuts Flakes, Post Alpha Bits, etc., and all the boxes have part of their names covered, such as "Rap Ut." It's pretty funny considering that TAGS cast did commercials for Post Grape Nuts, and other Post foods.
Andy's Rival - S6-E2
Other mistake: After Helen calls Andy to cancel their date, it cuts to Goober and Andy walking down the street and there's a brick wall adjacent to the grocery/meat store which should not exist.
Other mistake: When John Canfield's talking to Floyd, Andy, and the other men they're in front of the display window, which doesn't have any panes of glass.
Other mistake: While Andy and Helen are rearranging everything in the squad car's trunk to make it all fit, Helen picks up the large thermos and written on its bottom we see "Reggie personal." Reggie Smith was property master on set. (00:12:45)
Other mistake: During the scenes at Goober's filling station we can see a few brand names covered. For example, the "Chew Mail Pouch Tobacco" wall thermometer sign has its brand name "Mail Pouch" covered with black tape, and all the yellow "Pennzoil" cans have half the letters of their brand name painted yellow, so all the cans read "en zo."
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.
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)
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.