Which character is the most heroic, and what ultimately does the story say about the ideal of heroism?

This is a challenging question because Modernism complicates how we define heroism, and Faulkner tends to be rather tricky in how he presents character. In a traditional sense in which the hero sacrifices personal interest for the greater good, Cash seems to be the obvious choice for the novel's hero. He dutifully makes Addie's coffin, allowing her to watch him make sure it is a well-made casket but also one in which the workmanship is beautiful (chapter 18). In his quiet and methodical way, this seems to be Cash's way of expressing his relationship to his mother. Similarly, he risks himself in trying to get the wagon bearing the family and the coffin across a flooding river. Breaking his leg a second time, Cash quietly suffers the pain and the then the indignity of his father's inept approach to dealing with the broken leg. Cash is the model of stoic duty. This is likely misplaced, given the quality of the family and the cause to which Cash dedicates himself. On the outside, he may present himself as a Hemingway-esque hero, though nothing in this novel rises to the level of the ideals Hemingway would engage in his novels.
If we see the novel as a parody of the Odyssey, then the goal is to bury Addie in her home county and to successfully return home. As a comedy and a quest story, self-interest and cleverness in outwitting others to overcome obstacles would be the higher value. In this case, we see most of the characters seeking their own interests apart from the group. Here, we would have to see that Anse is the comic hero, for he gets what he wants (new teeth) as well as a new wife, and the completion of his quest cost him nothing. He had already lost the respect of his children and acquaintances, and he seems unbothered by the distress his children experience.
Placing these two very different characters in contrast, it seems that the message about heroism is clear. Faulkner seems unimpressed, or unbelieving, of conventional notions of masculine heroism. This novel should be read as a dark and ironic comedy; as any other type of story, it is too dark to enjoy. Faulkner's portrayal of human character—from the farmers who lend their help to the family on the foolish journey but do nothing to protect the children from these parents to the store clerks who rape Dewey Dell, to the minister who fathers the illegitimate Jewel, to each of the characters in the Bundren family—is bleak and best appreciated in terms of the ability to survive. Anse, who survives without doing any sweat-inducing work, who is able to remarry within days of his first wife's death, is able to use his children to further his own interests.