Lonesome is quite the character. He does move through a journey, but it certainly doesn't align with Campbell's definition of a hero. Maybe more along the lines of Aristotle's tragic hero?
Can an immoral man go on a hero's journey?
It is quite possible in literature for a hero to begin as a hero. It is only post-Byronic heroes that are dubious in their heroic qualities, and of course this Byronic tendency only increased over time, especially as a result of Modernism and Postmodernism. Made prominent by Byron, this type of hero does not posses high standards of heroic virtue but does possess dark qualities and rebellious qualities. Campbell expands this and redefines the heroic a bit as one who responds to "these pulls and pushes as we journey through the story that is our life," which still aligns his definition with the Byronic type. What, by these definitions, makes Rhodes a hero is his anticipated (by Miller) phoenix-like rise from betrayal to restored success.
I think even the most immoral of characters can become a hero. One novel that clearly shows this is The Power and the Glory by Graham Greene, whose protagonist, who is a very unsavoury and dislikable individual, actually shows himself to be a true hero at the end of it in spite of his many faults and problems.
I think the point you brought up about Aristotle's tragic hero is very relevant when answering this question.
The tragic flaw is important in the journey of the hero, according to Aristotle and others who adhere to his theologies.
An immoral man can take on a hero's journey. He may not know it is the journey of a hero at the beginning, but (like liteacher states) "no hero beings as a hero." In the end, it is the growth of the character which counts the most.
That's an interesting question. Yes, I think an immoral man can go on a hero's journey. He should collect some morals along the way. There are many instances of reluctant heroes who are pushed into the journey by some outside force and actually change into better people. No hero begins as a hero, and they all undergo a transformation.