Is it permissible for a woman to read the Quran without a headscarf, or without ablution, or during menstruation?

A woman may recite the Qur’an without a head covering, unless she is in a place where non-mahram men can see her, in which case she must cover her entire body. The same applies if she is praying; she must cover her entire body except for her face and hands. She is also permitted to recite the Qur’an without ablution, or while menstruating or in postpartum bleeding, but she must not touch the mus'haf unless she is in a state of purity.