It is strange that humans in the Garden of Eden are tempted by a snake. The strangest part, of course, is imagining how an animal could approach humans as an apparent equal. (I plan to partly address that question in a future post.) Another interesting question is: why a snake? Why not a different animal?
To possibly answer this, let's look at some background information about the sin in the Garden of Eden:
1. Eating from the tree makes humans aware of the their nakedness, and the first baby is born afterwards
2. The tree of “knowledge” may be related to “knowledge in a biblical sense”
3. The “fruit” may allude to “being fruitful and multiplying”
4. The fruit is described as very attractive looking ("a desire to the eyes, and beautiful to perceive"), which is why the woman has trouble keeping herself from eating it.
5. The woman's punishment apparently relates to her desires and their consequences.
It is clear from all this that the sin is very closely related to sexuality.
With that background, let me mention the following idea:
6. In popular culture, the word “snake” is sometimes used as a euphemism for a body part.
As support for this interpretation, note that 1) The woman is tempted by the snake, not the man. 2) The snake begins the story standing (which real snakes can't do) and is punished by no longer being able to stand.
So the image of the snake may be chosen to further develop the role of sexuality in the story. And this not the place to elaborate.
The midrash that the snake wanted to mate with the woman probably flows from these considerations (as well as the fact that the man too tried mating with animals).
In order to uphold my reputation, please forget that I ever said what I just said. But I do think this idea explains some things pretty well. Even if you can't repeat it at the Shabbat table.