Was the questioner's action correct in completing the prayer aloud after the time for Fajr had ended? And is he sinful for not checking his clothes and making sure of the presence of madhy (pre-seminal fluid), even though he repeatedly sees madhy after prayer?

If you wake up and your private part is erect, check your clothes before praying, as it is possible for something to have exited. The fact that you have not found anything on some occasions does not mean negligence. If you wash your private part and remove the madhy (pre-seminal fluid) and repeat the prayer, what you have done is correct, and your prayer is considered a qada' (making up a missed prayer), based on the saying of the Prophet, peace be upon him: "Whoever catches one rak'ah of the prayer has caught the prayer." The preponderant opinion is that the Fajr prayer, when made up, should be recited audibly.