The Question and Answer section for Sir Gawain and the Green Knight is a great resource to ask questions, find answers, and discuss the novel.
There really isn't a romance. Lady Bertilak tries to tempt Gawain into a romantic relationship but this is only to test his chivalry.
D. King Arthur believes in treating all people fairly, no matter how they treat him.
Gawain's most endearing and overriding trait is his honesty. The fact that he is shamed by his own actions, which for most of us are human qualities, makes him all the more endearing.