Is the method followed for making up missed prayers—that is, praying the fard of the current day, then the supererogatory prayers, then the missed fard—correct?

Missed prayers must be made up. If their number is known, all of them must be made up. If unknown, the making up continues until one is reasonably certain of having fulfilled their obligation. The making up should be done in a way that does not harm the body or livelihood. What a person feels of delusion or self-admiration is among the stratagems of Satan to discourage them from worship. Therefore, one must strive against the self and sincerely devote acts of obedience to Allah.