Hoolihan is gone over in detail in thread 14070, with definitions from knowledgeable people like Ramon Adams ("Western Words") and J. E. Lighter ("Historical Dictionary of American Slang") as well as those who are just guessing. houlihan Old Paint

One problem is whether hooleyann, the roping term, hoolihaning, bulldogging term (practice barred at rodeos now), defined in 1933 by Allen in "Cowboy Lore", or hoolihan (to have a good time) is meant. The words are easily confused and in part seem to be interchangable.
The three terms seem to be fairly recent (after 1900) coinage.
The word hoolihan (houlihan) shows up in the song in Sandburg (1927), Larkin (1931), Lomax and Lomax 1934. It may have surfaced in one of the recordings made in the 1920's, but I don't have them.
It does not appear in "Cowboy Songs," John Lomax, 1910 or 2nd ed. of 1916; not until the 1934 "American Ballads and Folk Songs" by Lomax and Lomax does it appear in their volumes. He (John) said he got the song from Boothe Merrill in 1910 at Cheyenne's 'Frontier Days', but I doubt that it contained the verse with 'throw the hoolihan.' (I remember the word in New Mexico in the late 1930's; to have a party).

Larkin (from Lynn Riggs), Sandburg and Woody Guthrie made the 'fiery and snuffy' verse popular; its first appearance is in Sandburg, 1927, "An American Songbag." It also has been gone over in the thread cited above; see posts by Lighter and Richardw in thread linked above. I'll take Richardw's word for the meaning. However, other westerners say 'fiery' is another name for paint, and snuffy means a snuff, or buff-colored horse. Others say it means lightning and thunder, or horses that snort and spook, tending to buck and go.
Charley, take your pick.

The version "I Ride an Old Paint" is not a 'music hall' song; it never appeared in a musical. Lynn Riggs, who gave it to Margaret Larkin, was from a cattle ranching family. Whether Riggs wrote it or got it from "a buckaroo," however, can be questioned (see comments by Lighter, thread 14070); it may not a valid song of the range, but it is just as popular among cowboy singers as "Old Paint" sensu stricto.