What do the various actions of Gawain (good and bad) tell us about his character? Is he still a hero because he gave in to a weakness?
Gawain is a hero because he gave in to a weakness--and recognized it, and asked forgiveness for it, and took measures to ensure it didn't happen again. He is humble in his contrition, but those around him recognize his sincerity. The Green Knight (Bartilak) makes plans to break bread with him again the next year, and his fellow knights are moved to join him in his contrition by wearing similar green scarves/girdles/sashes. Gawain is a hero worth emulating.
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