What is the ruling on making up missed prayers, and is it permissible to compensate for them by performing extra supererogatory prayers? And what is the number of rak'ahs required to make up, for example, the Fajr prayer?

Fasting prayers must be made up according to the majority of scholars. As for one who intentionally leaves a prayer until its time has expired, some scholars believe that he can never make it up, and that he must instead increase in good deeds, voluntary prayers, and seeking forgiveness. This opinion has been chosen by a group of early and later scholars, such as Sheikh Al-Islam Ibn Taymiyyah, Ibn Hazm, Ibn Al-Qayyim, and Ibn Uthaymeen. They did not specify a certain number of rak'ahs or supererogatory prayers that would substitute for a single obligatory prayer.