Trivia: In Walker's Drugstore, among all the magazines by the wall there's a TV Guide magazine (from Oct 9, 1954) on the shelf, and on its cover is Lucille Ball the co-owner of Desilu Productions - which produced TAGS, and also filmed at Desilu Studios.
Trivia: Howard McNear who played Floyd The Barber was the barber in Leave It To Beaver, Season 2, The Shave.
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.