So is Jimmie Driftwood recognized as the songwriter, or is it someone else (e.g., Don Warden).
Who holds the copyright on the lyrics? Was the tune composed in 1956 or is it borrowed or adapted from an older folk tune?

It sounds like there is still some dispute.

Here's what Wikipedia says about the song's origins. (Of course, that could change in a few minutes.)

"The melody has its roots in a well-known American fiddle tune "The 8th of January", which was the date of the Battle of New Orleans. Jimmie Driftwood, a school principal in Arkansas with a passion for history, set a historical account of the battle to this music in an attempt to get students interested in learning history. It worked, and Driftwood became well known in the region for his historical songs. He was "discovered" in the late 1950s by Don Warden, and eventually signed to a recording contract by RCA, for whom he recorded 12 songs in 1958, including "The Battle of New Orleans"."