I would agree with post 3. Oedipus is heroic in his determination to find King Laius' murderer to relieve the issues from the plague. Oedipus vows that the murderer will be exiled, and, when he learns that he, himself, is the murderer, he follows through with his vow to punish the murderer. Oedipus lost everything, but he does not seem to blame others in the end. He commits to being punished as if he had intentionally tried to make the terrible prophecy about his life come true.
I think that a strong case can be made that Oedipus is a hero. If we accept the premise, that a hero "fights for a good cause," the opening of the drama is reflective of this. Oedipus commits all of his efforts and resources to helping his people. Oedipus himself is secure and clear with his rule. Yet, he extends himself at the need of his people. He is a hero in this, as he proves to be an example of how political rule should be as opposed to how it is. Another heroic quality of Oedipus is his refusal to stop in his quest to find the truth and to help his people. Jocasta, his mother/ wife, tells him to stop, praying to Apollo that he will stop. Yet, he does not. He gains nothing from this except the alleviation of his people's suffering. This is heroic.
It is at the ending, though, where Oedipus displays his greatest heroism. He is confronted with the most awful of realities. The awareness to self is brutal. He has married his mother, killed his father, and his children are really his siblings. His mother/ wife has hanged herself. He has blinded himself. Yet, at the end, he begs the people to have mercy on his children. At the worst and lowest point of Oedipus' life, he demonstrates heroism by asking for mercy on others, his children. He shows absolute heroism in not being self- absorbed nor seeking to find a coward's way out of his predicament. What he does in blinding himself and consigning himself to exile is heroic in the way it seeks to right a wrong. Oedipus is heroic in his desire to restore moral order and a just structure to a machine that he unkowingly broke. I don't think he can be considered a coward. I think he has to be seen as heroic with this in mind. The fact that he does not wish to become a hero at play's end is precisely what makes him heroic.
The question is wrong.
The right question would be: "Is Oedipus a brave man, or a coward one ?"
He can't be a hero. Heros often fight for a good cause, from the perspective of the people he belongs.
But yes, Oedipus is a brave man. First, because he won the battle with his father, not knowing with who he is fighting with, but mostly because when he found out the truth about his acts, he got the power to punish himself.
So, he's not only brave but with a strong ethic feeling. He also prooved his wisdom when guessing the riddle.