Opie's Ill-Gotten Gain - S4-E8
Continuity mistake: Andy reads the inside cover of Barney's old schoolbook which has Barney's name "Bernard P. Fife" written out many times in it, but in two previous episodes Barney's middle name is either Oliver or Milton.
Opie's Ill-Gotten Gain - S4-E8
Continuity mistake: When Opie's surprised by the bike, his first closeup is a flipped shot - note the bell's on the wrong side of the handlebar, etc.
Opie's Ill-Gotten Gain - S4-E8
Continuity mistake: When Barney is "reciting" the preamble to the Constitution, he messes up his hair. When they show him in close-ups it is messed up, but when it goes back to him and Andy in the scene it is combed. It does that a couple of times.
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.