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.
What is the Iddah period upon death of husband? What is the ruling when the woman observing Iddah after death of husband leaves her home to visit relatives although her Iddah hasn`t ended? What is the ruling on her wearing gold during Iddah period?
All perfect praise be to Allah the Lord of the Worlds. May His peace and blessings be upon our Prophet Mohammad and upon all his family and companions.
For a woman whose husband has died, the 'Iddah* is four months and ten days after the death of her husband. If a woman is pregnant, the 'Iddah lasts until she gives birth. Moreover, she has to mourn, not wear gold, perfume nor saffron-colored garment. The evidence on this is that The Prophet (PBUH) said: "It is not lawful for a Muslim woman who believes in Allah and the Last Day to mourn for more than three days, except for her husband, for whom she should mourn for four months and ten days." [Agreed upon]. And Allah The Almighty Knows Best.
*The iddah is a waiting period that a Muslim woman observes after the death of her husband or after a divorce. The Quran says: For those men who die amongst you and leave behind wives, they (the wives) must confine themselves (spend iddah) for four months and ten days.
Is it a condition that the meat of the Eid sacrifice should be distributed in order for the sacrifice to be accepted by Allah?
The poor and needy should be given from that sacrifice , even if it be a small portion,(One kilo for example) as raw meat so that they would cook it the way they want. The remainder of that sacrifice is either distributed, eaten, or offered to people in a banquette .