A man insulted the Divine Essence; is it obligatory for him to perform Ghusl?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
Insulting or blaspheming the Divine Essence (Dhat al-Ilahiyyah) constitutes apostasy (Riddah). The perpetrator of this grave sin must immediately repeat the two testimonies of faith (Shahadah) and sincerely repent to Allah the Exalted. It is recommended (Sunnah) for them to perform a ritual bath (Ghusl), though it is not a mandatory condition [for the validity of their return to Islam]. And Allah the Exalted knows best.
Is it permissible for a woman to sit with her brothers-in-law?
It is impermissible for a woman to sit with her brothers-in-law, and it is also impermissible for a person to have a Khulwa (seclusion) with his sister-in-law.
Is it permissible to slaughter a sheep on Eid al-Adha with the intention of charity (sadaqah) on behalf of a sick person, and with the intention of udhiyah (sacrificial animal) at the same time?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
The udhiyah (sacrificial animal) is an act of worship intended for its own sake; therefore, it must be slaughtered with the intention (niyyah) of udhiyah.
Whoever intends to draw closer to Allah through this act of obedience with the intention of udhiyah, and also intends to gift the reward to another person, there is no Islamic objection to that. And Allah Almighty knows best.
Is it permissible for women to gather in one of their homes to pray Tarawih?
Yes, it is recommended (mustahabb) for women to pray Tarawih in congregation if they gather, even if it is not in the mosque.
In fact, praying at home is better for them, as it avoids the potential concerns associated with going to the mosque.