After I had prayed, I found a drop of menstrual blood and was unsure when that happened; what is the ruling on my prayer and fast?
All praise is due to Allah, peace and blessing be upon the Messenger of Allah.
Your prayer and fast are correct ,and you should not make up for them, and we ask Allah, The Almighty, to duely reward you for these acts of worship. Your menses start once you see blood because the situation remains as such unless there is certainty or predominant conjecture, as per the Fiqh maxim: "Certainty is not removed by doubt". And Allah Knows Best.
Is it permissible for someone to make a supplication to Allah to marry an already married, or an engaged woman?
No, it isn`t, and the person who does so is sinful before Allah, The Almighty.
Is the father a Mahram (Non-marriageable) to his son`s mother-in-law?
The father is a non-Mahram (Marriageable) to his son`s mother-in-law, so it is impermissible for them to look at each other, or to have a seclusion (Khalwah).
I have asked a marriage official to conclude my marriage since I have no proxy and, being an adult, I can act as the proxy of myself, but he refused. What should I do?
All perfect praise be to Allah the Lord of the Worlds. May Allah`s peace and blessings be upon our Prophet Mohammad and upon all his family and companions.
In order for a marriage to be valid there must be a Wali (Guardian) for the woman: father, brother, paternal uncle or any paternal relative. If there isn`t any, then the judge can act as the woman`s guardian and conclude the marriage contract. And Allah The Almighty Knows Best.