I know that "Kansas City" was written by Lieber & Stoller, darlings of the Brill Building (the place with the "gold plated door" that Gram Parsons mentioned). The Coasters were outlets for Lieber & Stoller so assume the two songs by the Coasters are also by them. Rock 'n' Roll? More like Tin Pan Alley.

"Running Bear" and the "Battle of New Orleans" are really novelty songs, and "What'd I Say" is pure Rhythm 'n' Blues"

Not that they aren't good songs, just that they ain't Rock 'n' Roll.