Is the father a Mahram (Non-marriageable) to his son`s mother-in-law?
The father is a non-Mahram (Marriageable) to his son`s mother-in-law, so it is impermissible for them to look at each other, or to have a seclusion (Khalwah).
Which takes precedence: the 'aqīqah or the uḍḥiyyah?
All praise is due to Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
The 'aqīqah is an act of worship through which a servant draws closer to Allah the Almighty in gratitude for the blessing of a newborn child. It is a confirmed Sunnah for those who are financially capable, and Allah does not burden any soul beyond what it can bear.
However, the uḍḥiyyah takes precedence, as it is a confirmed Sunnah established from the Prophet ﷺ through stronger and more numerous narrations — indeed, the Ḥanafī scholars hold it to be obligatory. Furthermore, its time is narrow and limited, expiring with the passing of the days of Eid, whereas the 'aqīqah may be delayed until one becomes financially capable of performing it.
And Allah Almighty knows best.
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."
I have a brother who is harsh in his dealings [with me], and many problems have occurred between us, and I do not intend to reconcile with him. What is the ruling of Islamic Law on that?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah.
It is not permissible for a Muslim to abandon or boycott his fellow Muslim brother for more than three days. This is based on the saying of the Messenger of Allah, peace and blessings be upon him: "It is not lawful for a Muslim to forsake his brother for more than three [days], meeting each other but one turns away and the other turns away, and the better of the two is the one who initiates the greeting (salam)." (Reported by al-Bukhari.)
A Muslim must strive to end such estrangement (hajr), so as not to fall under the prohibition mentioned in the hadith. If he takes the initiative to reconcile but the other party does not respond, then there is no sin upon him. We remind [ourselves] of the saying of Allah, the Exalted: "And let them pardon and overlook. Do you not wish that Allah should forgive you? And Allah is Forgiving and Merciful." (An-Nur/22). And Allah, the Exalted, knows best.