How is the number of days of a menstruating woman's period calculated, and is the day counted in full if menstruation began in the middle of it? And how are the missed or obligatory prayer times determined when menstruation or purity begins at different parts of the day?

The menstrual cycle is calculated by the onset and cessation of bleeding, not by the number of days. When a woman sees blood, she must stop praying, and when she sees purity (cessation of bleeding), she must perform ghusl (ritual bath) and pray. If she menstruates before an obligatory prayer, she is not required to make it up according to the Maliki and Hanafi schools, and this is the preferred opinion. However, if she becomes pure before the time for the prayer expires, she must perform it.

Regarding fasting, if she menstruates even for a moment before sunset, the fast of that day is invalidated, and she must make it up. If she becomes pure during the day, she is not required to abstain (from eating and drinking), and she must make up that day.