From my own experience working with horses, a houlian is a loop used for catching horses milling around in a corral. It's done with one easy swing to the side and an underhand toss. The loop travels vertical to the ground, and the horse basically runs through it. This, for the reasons that Les posted earlier. It does appear that the cowboy is using it as a slang expresion for having a good time. I don't know how the two are connected. From the context, I had always pictured Old Dan as a pack horse. (Ride one, pack one.) And, to us, firey, and snuffy have always meant hot blooded, nervous, or otherwise agitated horses or cattle. I've always loved this song.
Take care
Timehiker