Character mistake: In the first scene of the show, Andy, Opie and the mayor are on a pier fishing a lake. They are casting with deep sea reels. If you've ever tried casting with a deep sea reel, it doesn't work out well. You can see there no equipment on the poles.
Character mistake: When Andy and Barney are looking through the yearbook, Barny's middle name is Milton. But Barney says many times throughout the series that his name is Bernard P. Fife.
Character mistake: Mr Darling says there are only 5 strings on the guitar he hands to Andy. You can see all 6 strings when the camera shows Andy playing Dooley.
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.