Do certain religious traditions believe that Eve had sex with the Serpent?

Question

On p.169 of my edition of de Beauvoir's The Second Sex, she includes a footnote remarking

The moon is a source of fertility; it is seen as the "master of women"; it is often believed that the moon, in the form of a man or a snake, couples with women. The snake is an epiphany of the moon;...it watches over holy sources, the Tree of Life, the Fountain of Youth, and so on, but it is also the snake that takes immortality away from man. [To reiterate] It is said that it couples with women. Persian and rabbinical traditions claim that menstruation is due to the first woman's intercourse with the snake.

Is this true (i.e., do Persian and rabbinic traditions actually say this)?

This question is (probably) unanswerable in the affirmative without quoting a religious source. Hence, if rabbinic tradition does say as much, I would prefer an answer that provides a (brief) summary of the reasoning that led to such a claim.

_{Since I'm citing a print text, some bibliographic data:
My book is the 1st Vintage Ed., May 2011, ISBN 978-0-307-27778-7, with translation done by Borde and Malovany-Chevallier. I am told other translations can differ dramatically in wording and content. p.169 lies a few pages in to Part III, Ch. 1 ("Myths").}

Answer 1

Eve having Sex with the serpent

This is also called the serpent seed theory.

The idea is that the arch angel Samael was the serpent and used that form to have sex with Eve, which then resulted in the birth of Cain.

As for the reasoning for the claim, see for example this article by Howard Schwartz:

This myth is a response to the enigmatic verse in which Eve says, I have gotten a man with the aid of Yahweh (Gen. 4:1). Targum Pseudo-Yonathan translates this verse as “I have acquired a man, the angel of the Lord.” One reading of this verse in the Talmud (B. Shab. 146a) suggests that Eve had intercourse with the serpent: “When the serpent consorted with Eve, he cast impurity into her.” This interpretation is echoed in the Zohar: “From the impurity with which the serpent infected Eve emerged Cain.” Pirkei de-Rabbi Eliezer builds on the talmudic interpretation, but changes it in an essential way. Here the true father of Cain is the angel Samael, who came to Eve riding on the serpent. Indeed, in this passage the angel and serpent are closely linked, creating a satanic figure and suggesting that Eve had intercourse with the serpent, a powerful phallic symbol.

[...]

In order to explain why Cain was evil and Abel was good, Zohar 1:54a explains that Cain was conceived from the side of unholiness and Abel from the side of holiness

The idea is rejected by mainstream Christianity, but is supported by the white supremacist Christian identity movement, which claims that only white people decent from Adam, while Jews (and other non-white people) decent from Cain who descends from the serpent.

Origin of Menstruation

The idea that menstruation is a punishment for Eves supposed misstep exists in Judaism.

See eg the Jewish Womens Archive:

The Talmud asserts that Eve was given ten curses, seven of which are learned from Gen. 3:16: “I will make most severe”—these are the two drops of blood that aggrieve a woman, the blood of menstruation and the blood of virginity;

For more in-depth information, see for example this book by Sharon Faye Koren. It mentions the myth of Eve having sex with the serpent, and menstruation resulting as punishment for it. It also mentions that this is not an idea that is prominently mentioned in the Talmud, but appears instead more prominently in Kabbalah.