Is it obligatory for someone who drank water and took medicine after the imsak time specified in the Ramadan calendar (4:00 AM) and ten minutes before the actual Fajr Adhan (knowing that the Adhan is at 4:15 AM) to make up that fast? And is their fast considered invalid?

A fasting person may eat and drink until the emergence of the true dawn (al-fajr al-sadiq), and not according to the imsakiyah (pre-dawn meal timetable); based on the Almighty's saying: "And eat and drink until the white thread of dawn becomes distinct to you from the black thread [of night] from the dawn." And based on the Prophet's (peace and blessings be upon him) saying: "Indeed, Bilal calls the Adhan at night, so eat and drink until Ibn Umm Maktum calls the Adhan." Thus, if the muezzin calls the Adhan at the emergence of dawn, then the fast is valid.