What role do gender and gender roles play in The Epic of Gilgamesh?

This question about gender and gender roles in
The Epic of Gilgamesh is challenging. In going about answering it,
I think that I would first focus on the key characters in the story, making note of both
their gender and their actions, and then look to see if any sorts of patterns emerge.
For at least some of the male characters, maleness seems to be tied to destruction or
harm: they’re pretty harsh or unkind or wild, break things, kill things, and so on. At
least some of the main female characters, by contrast, seem to have a civilizing or
nurturing influence (e.g. the prostitute or courtesan, who civilizes Enkidu, or
Utnapishtim's wife, who doesn’t want Gilgamesh to leave empty-handed), but they aren’t
always all too peace-loving, either (e.g. Ishtar turned past lovers into animals and
lobbies to have Gilgamesh killed).

To me, it doesn’t appear
that the epic is making a clear and permanent distinction between all men and all women,
but the epic does seem to put emphasis on pairings of humans and/or gods. There are a
number of references to couples (gods, humans, and animals) throughout the story. Of
course, not all of the couples are heterosexual. Gilgamesh and Enkidu are easily the
most prominent couple in the story.