1. Does the current calendar advance the time for Fajr by five minutes, meaning there is no problem in eating during the Adhan according to this calendar? And is the make-up fasting performed based on this confusion valid, or must it be redone? 2. How many extra days should be added when making up for missed Ramadan days due to a legitimate excuse, knowing that your monthly period ranges between six and seven days, but making up the fast might extend to eight days?

It is not possible to determine the accuracy of the Fajr adhan timing in your calendar. You should refer to the scholars in your country if you cannot ascertain the beginning of the time through the Sharia signs.

If you drank during the adhan believing that Fajr had not yet broken, your fast is valid. Shaykh Ibn Uthaymeen stated that whoever eats, thinking that Fajr has not yet broken, and then it becomes clear that it has, their fast is valid. Some scholars have permitted eating or drinking for a minute or so after the adhan if the muezzin relies on the calendar. Shaykh Al-Fawzan said: "The calendar is a sign to be used, and one should not delay significantly beyond its limit. As for a minute or two, there is no harm." Regarding your doubt about the number of days for make-up fasts, you must investigate and act upon your predominant assumption, and fast what you predominantly believe will absolve your responsibility.