Is a woman to remain in her state of Ihram, and is expiation (fidyah) obligatory upon her after committing prohibitions of Ihram, such as cutting (hair or nails), applying perfume, and combing hair, out of her ignorance of remaining in Ihram after the onset of menstruation during Tawaf, and how does she complete her Umrah and exit from it?

Your circumambulation is invalid due to the onset of menstruation during it. You remain in your state of ihram, and you must refrain from the prohibitions of ihram. Then, you should return to perform the Umrah, by performing the circumambulation, the sa'i, and then shortening your hair. With this, your Umrah will be complete. What you committed regarding the prohibitions of ihram due to ignorance of the ruling incurs no penalty. Some scholars are of the opinion that a fidyah is obligatory for every prohibition that involves destruction, which is fasting for three days, or feeding six poor persons, or sacrificing a sheep. You have the choice between these. Repeating the prohibition before offering the fidyah only necessitates one fidyah.