Is Walter Mitty a hero and, if so, why?
I would answer that he is not a hero. In twentieth (twenty-first?) centruy term, hero is often defined by what a person a does: a person must act to have meaning in the world. Such a person who acts and accepts responsibility for his actions is called "authentic." This sense of heroism derives from a philosophy called "existentialism." In any case, Mitty does not act; he does not take full responsibility for the worth of his life. His mother, who "hen-pecks" him, ultimately maintains control over him and he is unable to confront her and live his life on his own terms, fully and completely. Unable to confront her, he instead retreats, goes back to the life of a child almost, living in day dreams. Yes, he manages to support his family and so on, but as for being an authentic human being, he falls short of the mark. Indeed, "Walter Mitty" has become a general term that refers to such a person--the kind of guy who would rather play out life in his fantasies rather than confront it and live it to the fullest.
Walter Mitty spends most of his adult life living through his daydreams. He feels trapped in his world, by his job that he doesn't like and by his wife that is characterized as demanding. In those dreams, Mitty is the stereotypical hero, smart, Witty and in control. He saves every situation. However, in his life, he is unwilling to express how he feels and is often failing at his responsibilities because he is so lost in his dreams. In this sense, he is not a hero, but a sad and unfulfilled man.
Others may argue that Mitty's daydreams are his way of being content with the day to day boredom and monotony of daily life, the things we all have to "do" in order to fulfill our roles in our world. In this case, Mitty is a hero, because he has found a way to have happiness and remain a husband and employee at the same time.
Walter Mitty is an example of an antihero; i.e., a protagonist who lacks the expected qualities of the hero. An antihero may perform heroic acts, but usually they are for his or her own benefit. Mitty is a hero in his mind only. Enotes.com has several good articles describing the antihero in literature; just search for the keyword antihero. I've inserted a link to Wikipedia also.