What is the ruling on making up missed prayers during prohibited times?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
It is permissible to make up (qada’) missed prayers at any time, even during the periods when prayer is generally prohibited. The prayers that are forbidden and considered invalid during these times are 'absolute voluntary prayers' (nafl mutlaq)—which have no specific cause—and voluntary prayers whose cause follows the prayer itself, such as the Sunnah of entering Ihram or the Sunnah of the Istikharah prayer. Furthermore, no prayer is considered disliked (makruh) during these prohibited times when performed within the Meccan Sanctuary (Makkah al-Mukarramah).
It is stated in Bushra al-Karim (Vol.1/P.181), one of the Shafi’i texts: 'It is not forbidden to perform prayers that have a cause that is not delayed (i.e., the cause is preceding), such as making up a missed prayer (fa’itah)—even if it was a voluntary one—and the funeral prayer (janazah); or a cause that is simultaneous, such as the prayer for rain (istisqa’) or the eclipse prayer (kusuf)... and the Sunnah of wudu, the greeting of the mosque (tahiyyat al-masjid), the Sunnah of circumambulation (tawaf), the Sunnah of arrival, and the prostrations of recitation (tilawah) or thankfulness (shukr). These mentioned prayers and their like are not forbidden provided that one does not specifically intend (ta'ammud) to perform them during the disliked time because it is a disliked time. If one does so intentionally, it becomes forbidden, even if it is a mandatory makeup prayer that is due immediately; because in that case, one is acting in defiance of the Sharia. This is in contrast to when one does not specifically seek out that time, even if the prayer happens to fall within it, or if one seeks it for another purpose—such as delaying a funeral prayer to that time so that a larger number of people may pray over the deceased; in such cases, it is permissible and valid... And it is forbidden to perform prayers with no cause at all, like absolute nafl, or those with a delayed cause, such as the Istikharah prayer, the prayer for Ihram, the prayer for a need (hajah), the prayer before leaving the house, or the prayer before execution; because their causes occur after the prayer itself.' And Allah the Exalted knows best."
What is the ruling on the intention of fasting, where is its place, and when is its time?
The intention is a pillar of acts of worship; it is essential, and worship is not valid without it.
Its place is in the heart, and verbalizing it is recommended so that the tongue reminds the heart. Its meaning is to be determined to abstain from nullifiers of fasting during the upcoming day with the intention of worship and obedience to Allah Almighty. This meaning is naturally present in every Muslim on every night of Ramadan, so there is no need for obsessive doubts. If one says, "I intend to fast tomorrow for the sake of Allah," it removes any doubts.
Its time is from after sunset until before the true dawn.
Is it permissible to offer prayer at home, or should it be offered in the mosque?
All perfect praise be to Allah, The Lord of The Worlds. It isn`t preferable for the man to pray at home as praying in the mosque is twenty seven times more rewarding. Therefore, this should motivate him to offer prayers in the mosque. And Allah 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.