How is Gawain connected to Lady Bertilak, Guinevere, and Morgan le Fay?
Gawain and Guinevere are connected only because of the split that occurs with Arthur's ranks (the knights) when Guinevere and Lancelot have an adulterous affair. Gawain is known for being a true and honorable knight. He takes his chivalric vow very seriously, which includes protecting women, children, the weak, and the less fortunate.
...when [Gawain's] brothers Agravain and Mordred plot to destroy Lancelot and Guinevere by exposing their love affair, Gawain tries to stop them. When Guinevere is sentenced to burn at the stake and Arthur deploys his best knights to guard the execution, Gawain nobly refuses to take part in the deed even though his brothers will be there. But when Lancelot returns to rescue Guinevere, a battle between Lancelot's and Arthur's knights ensues and Gawain's brothers, except for Mordred, are killed.
Lady Bertilak tries to seduce Gawain when he stays at their castle, just prior to his meeting with the Green Knight at the Green Chapel. Her husband Bertilak (who is also the Green Knight—he is a shape-shifter like Morgan le Fay) has asked his wife to test Gawain's honor and honesty by attempting to seduce him. She tries three times and each time he courteously resists her. All that she does her husband knows about, and for the most part, Gawain is a true and honorable man, as the Arthurian knights are reported to be.
Morgan le Fay is a popular figure in Arthurian legends, however, she also has a connection to the story of Sir Gawain and the Green Knight. In some versions of the tale, when the Green Knight reveals that he is really Bertilak at the end of the story, he admits that Morgan le Fay put him up to this test to see if Gawain was as honorable as he seemed to be. In this version, this news leads Gawain to speak of how deceitful some women can be. In other versions, Morgan is not mentioned, and Bertilak explains himself and makes peace with Gawain, who has been humbled by his experience.
In another well-known work—the anonymous late 14th Century poem 'Sir Gawain and the Green Knight' (the Green Knight was, like Morgan, a 'shape-shifter')—she was the instigator of the plot that began the story.
Additional source: http://www.enotes.com/topic/Morgan_le_Fay