Ride aronnd, little doney, makes more sense than Ride around little dogies. That line has always bothered me.
More than one singer has sung 'throw the hooleyan (not sure of their spellings); one was Johnny Cash. Not a cowboy, but through time he probably knew several and got the word from them.

The song has nothing that might suggest the rider was going to throw a big drunk other than the word, which originally could have been hooleyann (hoolian, other sp.)as Lighter suggests.

Adams, for houlihan, has the preferred meaning of a barred type of bulldogging in addition to 'painting the town red.' However, I remember the word houlihan used for a big party as far back as just before WW2 in New Mexico-Colorado, when lots of us quit whatever we were doing and started painting the town when war seemed inevitable and before the army caught us in 1941-1942. It has to be older thn that (I don't think any of us got it from Sandburg- no one I knew had heard of him at that time).