Never read such a law. While I'm not a legal-history expert, such a law would make no sense. It achieve no political objective or order (and you can bet that if it does, the Edo Bakufu will legislate it, like they did dress and seating). On the contrary it seems ripe to cause chaos. Not only that, what if the women were married? How could this kind of open infidelity be legal when successfully extending the clan onto the next generation was of the utmost importance?
Besides using the sash as a blanket making no sense, if someone wanted to rape they'd probably just do it on the ground.
I've read nothing to suggest pre-modern Japan had a different birth rate compared to any other pre-modern society.
Bullshit. Their clan name they take from their dad. Where else would they take them?
Now what is true that a huge number clan names (not all) came from place names, like the Suwa (Suwa lake, though technically the temples surrounding the late), Date (Date District), Uno and Mori after names of the family manors. But there are very simple and reasonable explanations for this. Since everyone trace their ancestors to a handful of important clans, it'd be super confusing if everyone was Minamoto or Taira or Fujiwara (they were on Kyōto court documents). Others, like newly-minted samurai who rose through the ranks from peasants, need family names to identify them. Still other peasants were ordered by law to take a family name during the Meiji Restoration.
These families got to take their name from somewhere, and place is one of the places where they did get it from.
The other source of this story is probably the system of marriage that existed since pre-history all the way up to mid-Heian Japan. In this system, the man would go to the residence of the woman, and call to her from the entrance or something. If she and the village elder or head of her clan agrees to the marriage, they would be married. Important to note here is that she has the right to reject, or at least her clan does. Once "married", they don't live together, and the child is raised by the mother's clan. "Divorce" is simply when the man stop coming to the woman's residence.
Yobai, the pre/extra-marital "visitations" at night continued into the early Meiji, mostly in rural and remote areas. But these are visitations, usually from the male to the female residence. It was not in the streets, and the female and/or her clan's (at least implicit) consent is needed. While under modern law we'd probably consider many to be some form or rape like date rape, there was certainly no law requiring the female to submit to any random man. I would also move to say that if anyone want to make fun of the Japanese for their promiscuity, their society likely isn't better (European bathhouses and brothels anyone?)