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 expiation for perjury?
One who commits perjury should repent, seek Allah`s forgiveness and offer an expiation which is: feeding ten indigent persons, or clothing them, or giving a slave his freedom, but if that is beyond his means then, he should fast for three days. Allah, The Almighty, Says in this regard (What means): "Allah will not call you to account for what is futile in your oaths, but He will call you to account for your deliberate oaths: for expiation, feed ten indigent persons, on a scale of the average for the food of your families; or clothe them; or give a slave his freedom. If that is beyond your means, fast for three days. That is the expiation for the oaths ye have sworn. But keep to your oaths. Thus doth Allah make clear to you His signs, that ye may be grateful." [Al-Ma`idah/89].
What should a mensurating woman, or in state of confinement do if she became ritually pure during the daytime of Ramadhan ?
If she became pure during the daytime of Ramadhaan, is desirable to complete her fasting during that day, and to make up for it along with other missed days due to menstruation. And Allah Knows Best.
What should a worshiper who has forgotten a pillar of prayer do?
If he/she remembered the forgotten pillar before offering it in the following Rakah (unit of prayer), then he/she should offer it, and continue the prayer, then offer Sujood As-Sahw (Prostration of forgetfulness) at the end of it. But, if he/she remembered that forgotten pillar after having offered it in the following Rakah, then he/she should offer a new Rakah in its (the one in which he forgot a pillar) place, and offer Sujood As-Sahw.