The banjo playing kid was child actor Billy Redden (at Wikipedia, or here, at IMDB), picked for the role. He seems to have only had other roles as a banjo player.

The music, well, I was watching re-runs of the Andy Griffith Show, with the Dillards acting as 'the Darlings' and was really surprised to hear them play Dueling Banjos. Turned out it "was written and recorded in 1955 as 'Feuding Banjos' by Country star Arthur "Guitar Boogie" Smith."

So none of this was impromptu. I wonder if the kid really played the banjo, but I think he might have.