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: 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.
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)
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)
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)
Aunt Bee's Restaurant - S7-E21
Visible crew/equipment: After Goober orders the chow mein dinner at the restaurant, in the following wideshot of everyone at the table, the curved chalk actor's mark is visible on the floor between Andy's and Helen's chairs. (00:12:20)
Aunt Bee's Restaurant - S7-E21
Continuity mistake: While Jack is trying to help Aunt Bee understand her fear, in a few of the closeups of Aunt Bee and Andy (seated on the couch) note the fruit bowl has moved and the candlesticks have vanished from the server, the large figurine on the left side of the hutch has vanished, and the flowers from the table have also disappeared. Then everything is back in their place. (00:21:10)
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)
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.