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.

Does the date for menstrual menses change, and how is the increase calculated?

If the bleeding lasts no less than (24) hours, but doesn’t exceed (15) days, and the purity interval between the two menses is no less than (15) days, then it is menstruation even if the date of menses changes, or its duration increases.

I can`t afford to get married, what should I do to curb my sexual drives?

You should offer a lot of voluntary fasting, keep busy with useful and permissible acts and make supplication to Allah, The Exalted.

Is the marriage, which lacks a valid legal contract, a guardian, and a court registration, valid?

It is incumbent that a valid marriage contract be concluded in the presence of a guardian and two trustful witnesses, and it should be registered in the court to protect the rights of the wife. Actually, a valid marriage contract is what differentiates between sound marriage and fornication.