In the novel of The Left Hand of Darkness, does the female gender actually have a role in the novel? Also, does it depend too heavily on the fantasy of a truly progressive league of worlds that respects cultural difference as expressions of the diversity of nature?

The Left Hand of Darkness is a 1969 science fiction novel by Ursula Le Guin. While it is not her first novel, and it is the one largely considered her first major success as a writer. It coincided with second-wave feminism, and, as it's primary setting is a remote planet full of androgynes (half men-half women). It was well-received upon publication.
The protagonist, Genly Ai, is an envoy from the planet Terra. He is on a mission (via interstellar ship that allows him to experience shortened time) to the planet Gethen, where he first tries to convince the king of the nation Karhide to join a confederation of planets, Ekumen. Gethen, in addition to being extremely cold, is populated by androgynous humans. Sexual intercourse is called kemmering, and its participants kemmerers (there being no appropriate or necessary name for males or females). Individuals play the male part and the female part in the same lifetime (resulting in a humorous scenario of a pregnant Gethenian King). Not having convinced the king, Genly Ai goes to the neighboring planet, Orgoreyn. This nation seems more agreeable, but is run by secret police who drug and imprison Genly Ai. He is rescued by former prime minister of Karhide, and his estranged friend, Estraven.
Ultimately, both nations join Ekumen, accomplishing Genly Ai's mission, but Estraven, pronounced an enemy of the state, is killed.
If this is a feminist novel, it is only in the sense that the planet Gethen is devoid of gender. Genly Ai, while shown sufficient hospitality at most major exchanges during his mission to Gethen, is marginalized for his perpetual maleness, by which all are fascinated, and which no one understands. The novel does not portray the androgynous race as superior. On the contrary, while little is shown of it first-hand in the course of the novel, it is suggested that Genly Ai's planet, Terra, is socially more advanced and first-world. Thus, Le Guin does not seem to suggest that gender equality solves all societal problems on the planet Gethen.
The novel does not rely too heavily on a utopian fantasy, insofar as science fiction is a genre in which, with respect to setting, nearly anything goes. Le Guin redeems the fantastical setting by discussing her characters' interior emotions and emotions surrounding interpersonal relationships (such as love, guilt, jealousy, betrayal, etc.).