Is the Fajr prayer caught if the time of sunrise enters while I am in the first rak'ah, and is the Athan application reliable for prayer times, and if the first prayer is valid, is the second considered a supererogatory prayer?

If you catch one takbira of the prayer before sunrise, you have performed it on time according to the majority of scholars. You can adopt this view as long as you suffer from waswas (obsessive doubts). Your prayer is valid and sufficient and does not become invalidated by the sun rising during its performance, according to the majority of scholars, including the Malikis. Only the Hanafis differ on this point. We advise you to turn away from these waswas and ignore them, as succumbing to them leads to great evil. The duration of your wudu (ablution) (25 minutes) is greatly exaggerated. Do not wash any limb more than three times. If you do that, your wudu will only take a short time.