Is it permissible for a menstruating woman to eat during the day in Ramadan? And are there any regulations for that?

A menstruating woman, a woman in post-natal bleeding, a traveler, and a sick person, if they become pure, arrive from travel, or are cured during the day in Ramadan, are not obligated to abstain (from eating and drinking). They are permitted to eat and drink, because their abstention would be of no benefit to them, as they are required to make up that day. It has been narrated from Ibn Mas'ud that he said: "Whoever eats at the beginning of the day, let him eat at its end."

As for the rule regarding showing one's breaking of the fast, the soundest opinion is that if the excused person's excuse is apparent, there is no harm in them breaking their fast openly. However, if their excuse is hidden, they should break their fast discreetly.