I punched the wrong key and probably 1000 words of carefully prepared commentary on "Old Paint" went up in smoke! Damn these machines!! But Mudcat wins because now I have to summarize....

John Lomax's claim that he first heard an "Old Paint" song at Cheyenne in 1910 (from his old college chum Boothe Merrill) is borne out by the fact that "Goodbye, Old Paint" first appears, not in the 1910 first edition of "Cowboy Songs" but in the 1916 revised edition.
The closest John and Alan came to publishing the song "I Ride an Old Paint," first printed by Sandburg in 1927, was when they boldly tacked all the stanzas of that song (with a couple o verbal alterations) onto the end of 1916's "Goodbye, Old Paint." Maybe a contributor sent them in anonymously sometime after 1916. Yeah, maybe.

Anyhow, this all means that the earliest known appearance of "throw the hoolihan" was not in 1910 but in 1927. (Sandburg spelled it "hoolian"; in her own book in 1931 Larkin spelled it "houlihan.")
That date makes it even less likely that the word was well-known on the trail as much as fifty years before. I like Q's suggestion that the word may well come from (nearby) Hooleyann Co., Texas rather than from the Irish surname "Hoolihan" or "Houlihan." eary info that this style of roping was somehow associated with that county would pretty much nail down the derivation; perhaps the same area was responsible for the "hoolihaning" of steers, as described by Louie Roy and other cow folks. We may never know, but the hypothesis haas the triple-threat appeal of culture, phonetics, and geography.

As Q also suggests, and as I am now persuaded by new research, "hoolihan" (however you spell it) is primarily a roping term, secondarily a bulldogging term. Ramon Adams's 1944 definition of "throw the hoolihan" as "paint the town red" now begins to seem like a somebody's SWAG to explain the phrase in - guess where? - "I Ride an Old Paint."

Without going into detail, let me say that the song makes the best sense to me when the words are rendered,

             A-ridin' old Paint, a-leadin' old Dan,
             I'm bound to Montan' for to throw the hoolian....

And,

             They're fiery and snuffy and rarin' to go.

That is, of course, to stompede.

IMO, the refrain makes little sense at is usually sung,

               Ride around, little dogies,
               Ride around them slow....
                  

I prefer to think the original word was not "dogies," but "Doney," another commonly given name for a horse, with the original meaning of "sweetie," as established, IIRC, in some far distant thread. The rider urges his horse to ride around the drowsy dogies slow so as not to spook them.

Woody Guthrie and the Almanac Singers seem to have added the stanza beginning "I've worked in the town and I've worked on the farm, And all I got to show is the muscle in my arm."   

Two interesting recordings of the other song, "Goodbye, Old Paint," are, first, the very influential one that Texas fiddler Jess Morris recorded for the Library of Congress about 1950 but said he learned from puncher Charley Willis before 1890 (Jess thought Charley had picked it up about 1870); and, second, a 2001 recording by Artie Morris - who says he's Charley's great-grandson.