Continuity mistake: Barney tries showing Andy what the "part bloodhound" dog can do for The Manhunt so Barney blows a dog whistle, and in the closeups of the growling dog the rope that was tied to his collar is suddenly lying on the floor, but it's back on his collar in the following shot.
Revealing mistake: When Barney blows the dog whistle to call Blue from the back room, the dog comes running and knocks Barney on his back, and we can see a bit of the cushion that's under Barney's back - it's visible within the shadow under him.
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.
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.