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.
Continuity mistake: In this episode, Ernest T writes Charlene a love letter. In a later episode when Ernest T goes back to school, it is revealed that he can neither read nor write.
Plot hole: When Barney and Andy are on the street watching the window painter, Andy looks over and says "I believe that's Briscoe Darlin's truck!" - but when they were driving up to his place the next morning, they both got out of the car and were wondering where their house was they heard music playing. The next scene shows them pulling in - seems that if he was able to recognize the truck in town, he would have been able to recognize the truck and they could have drove around looking until they found it?
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.