Other mistake: When Barney is ready to pour the plaster cast into the shoe prints, there are only a few prints on the ground. There should be a continuous set of prints from the barn door to wherever.
Factual error: We know from 2 episodes prior (Man in a Hurry) and generally mentioned throughout the show, the only filling station in Mayberry is Wally's Filling Station. In this episode Mayor Stoner is accused of only putting the new underpass in Mayberry because it will be next to Mayor Stoner's brother's filling station.
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.