Should I complete the surah after the Fatihah, or should I start the Fatihah anew if a worshiper joins me while I am praying alone? And should I complete a Sunnah prayer as a supererogatory prayer, or should I change it to an obligatory prayer if a worshiper joins me?

If you are praying alone and someone joins you while you are reciting a surah after Al-Fatiha, complete your recitation and do not repeat Al-Fatiha. The rak'ah counts for the follower even if they did not catch the recitation.

Al-Hajjawi said in Al-Iqna': "Whoever catches the bowing with him before he raises his head—without doubt as to having caught him bowing—has caught the rak'ah, even if he did not catch the tranquility with him, provided that he himself achieves tranquility."

You are not permitted to change a voluntary prayer into an obligatory one. Rather, you complete your prayer as it was. The most preponderant opinion is that it is permissible for someone performing an obligatory prayer to follow someone performing a voluntary prayer.