Are my prayers valid if I rely on the widely circulated calendar in my country, in which it is observed that Fajr al-Sadiq (the true dawn) occurs later than its adhan (call to prayer), and is the "Islamic Finder" program reliable in determining prayer times?

We have no knowledge of the accuracy of the aforementioned application. Such programs can be consulted but should not be relied upon unless they have been approved by trustworthy scholars. If Sheikh Al-Albani stated that the time for Fajr prayer is later than what is in the calendar, and the calendar is not officially approved, then his statement should be relied upon. It is difficult to verify this delay within the city due to the abundance of lights. It is advisable to consult the jurists of Medina to verify the time of Fajr, and to act according to their قول (statement/ruling) if they investigate Fajr outside the city and find it to be delayed.