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 me to eat from the animal that I slaughtered for Allah to protect my family?
It is permissible to eat from the non-vowed animal sacrifice, and the person is rewarded based on the amount of meat that he had given to the poor. However, there is no evidence in Islamic Sharia indicating that slaughtering an animal protects one`s family, but it is a way for thanking Allah, The Almighty, for his grace.
Is performing Umrah an obligation for who is capable. Is he accountable for not performing it in this case?
Umrah is an obligation once in a lifetime for every Muslim. When physical and financial ability is present, it is preferable to perform Umrah, as well as Hajj. If one can afford to perform Umrah but not Hajj, then they should begin with Umrah, especially since Hajj nowadays is not accessible to everyone who wishes to perform it. In contrast, Umrah is more open and easier to undertake. And Allah knows best.
Is it permissible to single out Fridays, Saturdays, and Sundays to make up for missed fasting without fasting a day before ,or after each ?
Yes, it is permissible to make up for missed fasting during the aforesaid days, and it is also valid to offer non-obligatory and vowed fasting as well. However, fasting a day before or after each isn`t obligatory. And Allah Knows Best.