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 to offer the Eid sacrifice in a non-Muslim country?
Eid sacrifice is permissible anywhere, and Islamic Sharia hasn`t determined that it should be in an Islamic country, but poor Muslims should be given a share in order for it to be valid since it is an act of worship. A portion of that Eid sacrifice should be given at least to one poor Muslim, and if that wasn`t possible then the one offering it should consume it along with his family.
I joined a savings club and was scheduled to receive the payout first. My friend asked to take my place in line, so I requested 50 dinars in exchange for letting her have my turn. What is the ruling on this?
This is impermissible, as it involves taking money without providing something of value in return according to Islamic law. And Allah The Almighty Knows Best.
Does the clipping of nails nullify ablution?
No, it does not, and it is preferable to wash the hands afterwards.