Reddit mentions of Jamaica Selects Jump Blues Strictly for You

Well, but lots of Fats Domino's stuff wasn't rock. It's r&b. And when he played rock, those records weren't particularly popular with Jamaican audiences. For a good (and really cheap!) look at what was popular in Jamaica check out Jumping The Shuffle Blues and its sequel Jamaica Selects Jump Blues Strictly For You

I don't think I've ever heard of New Orleans brass bands mentioned as an influence on Jamaicans, I doubt it would have factored very heavily though. They were, however, quite influenced by jazz, especially the current Bop sounds. Many of the musicians aspired to be jazz players, but for the most part knew they didn't have the chops. (That being said, Coxsone did put out a couple very enjoyable jazz albums on his Port O Jam label, I'm especially fond of "I Cover The Waterfront".) Those Jamaicans that did have the chops went on to very successful jazz careers (Joe Harriott (he pre-dates the ska era), Ernest Ranglin, and Monty Alexander in particular)