Generally any mulch consisting of organic material is good. Often organic mulch from palm parts is stringy and not as esthetically pleasing as pine straw, pine bark or wood chips. My favorite mulch is that produced by tree service chippers especially that of pine trees and broad leaf trees. It includes twigs, leaves, branches, bark and wood all homogenized in a nice mix. There is another problem with palm mulch — ganoderma zonatum, the dreaded butt rot disease. It is possible to bring the soil borne fungus to your site if the mulch contains debris from a palm that was removed because it was killed by ganoderma. So mulch consisting of palm debris has two strikes against it if used on another palm.
Regarding how far from the palm (or any tree for that matter) the mulch should be placed, I used to say 4 to 6 inches. But with wind, rain and mediocre landscape maintenance, that 6 inches quickly became zero inches. So now I say 18 inches which in reality turns out to be 12 inches.