Should one obey his mother even if she was wrong?
Obeying one`s mother is mandatory in matters that are beneficial to her and as long as she drive her childrennot to commit a sin.
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.
What is the ruling on having slight discharges of menstrual blood during a day of Ramadan, and before Maghrib time?
A woman who is certain that her menses have started during a day time of Ramadan is in a state of menstruation, and her fasting is considered broken at first sight of blood. However, she is rewarded for not breaking fast at the beginning of day time.
Is it permissible to shorten the eyebrows if they get too long?
Yes, it is permissible, but not desirable.