Just watched it again, a few more observations/thoughts. There was not blood currently coming from his nose. He has what looks like a makeshift patched hole in his cheek that is both bloody and appears to be leaking blood. Blood is coming out of his mouth. If he has been in jail a couple days considered "drunk" while he is supposed to be sobering up, his nose could have stopped bleeding. Chad calls the place a "fucking nuthouse" which the drunk repeats. Later in The Roadhouse one of the girls says "...Thats because your hanging out at the nuthouse...don't go in that nut place." Maybe it's a coincidence??? Also, the young woman describes seeing Billy as "so fucking scary....he had this look in his eyes...I started screaming...bloods gushing out like a waterfall...he looks and turns at us real strange..." In my interpretation, like the way the liquid was "water-falling" from the drunks mouth. Anyway, just some observations. You never know...but the nut house are references are making me think about Audreys predicament, and that she could possibly be in some sort of rehab.
There is definitely a 'symbolic' connection between the drunk and Billy(even if they are not the same person factually), and this also ties in with the 'blue rose' splitting into two, as seen in the two Coopers: the drunk is just repeating like Dougie, completely controlled by the environment around him and not even able to stop his mouth from dripping blood/oil(fire starter, pain, etc., head to 'wild west' where you get everything you want, the 'nuthouse', zebra, penguin, black and white extremes, etc., like the two drugged up prostitutes before); while Billy was jumping fences with a crazy look in his eyes, with extraordinary powers to violate the environment, much like Mr. C.
Another question is if this is factually the same Billy and Tina that Audrey was talking about. Is it really likely that the same Billy had seduced Audrey, the girl at the roadhouse making fun of the poor girl in end of 14, and 'Tina'? We know from before that there was another group of 'daters' who were engaging in relations structured in exactly the same way, except the names are changed (Clark is Billy, etc.); so that 'Billy' could just be a symbolic marker for a certain structural role that plays itself out repetitively throughout all dating relationships in the same way...and this Billy and Tina could be different people altogether, the drunk could be someone else, etc., but they all tie together through the roles they play....