Just found out about this manga so, this is a late comment.
most scholars consider touching the Quran during a woman's period forbidden (haram) with a few seeing it as that it's preferred to not do it (makruh), but all of them agree that they can touch it if they had a cover on it or something of the like, reading and reciting from their memories should not be spoken aloud too.
the demon spit isn't a real thing, just a superstition and a way to make children behave, we close it as a form of respect to the holy text.