What is the ruling on istibra' and how is it performed?

Istibra' (cleansing from urine) is recommended. It is for a person to ensure that nothing remains of urine in its passage before performing istinja'. Its method is: to clear the throat (or cough slightly), shake the penis (meaning to pass fingers along it to expel any remaining urine if needed), and walk a few steps to the extent that one thinks nothing of urine remains if walking is needed. Each person knows his own nature best. The preferred opinion is that this differs among people. The objective is for one to think that nothing remains in the urinary passage that he fears will exit later. And Allah the Almighty knows best.

Are school exams a valid excuse for breaking the fast in Ramadan?

School and university exams are not considered a valid excuse for breaking the fast, as most students take their exams while fasting without experiencing extreme hardship. Fasting does not conflict with exam preparation, and a Muslim seeks strength in obedience to Allah for both worldly and spiritual matters.

Who is the one required to slaughter the 'aqīqah?

Is a woman`s prayer considered invalid if non-Mahrams (marriageable men) saw her offering it?

A woman`s prayer isn`t invalidated if non-Mahrams saw her offering it, but she had better pray in isolation.