Is it valid to perform Tarawih prayer at home?
It is Sunnah for a Muslim to perform Tarawih prayer in congregation at the mosque.
However, if a person sometimes prays it at home in congregation with their family for a valid reason, there is no harm in doing so.
What is the ruling of Islamic Law on performing prayer on the upper floors of al-Masjid al-Ḥarām?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah ﷺ.
Facing the actual structure of the Ka‘bah (‘ayn al-Ka‘bah) is one of the conditions for the validity of the prayer. The scholars have clarified what is meant by "the actual structure of the Ka‘bah."
Imam ash-Sharwānī (raḥimahullāh) said: "What is meant by its ‘ayn (actual structure) is its physical body, or the airspace directly aligned with it — if the person praying is not inside it. But if he is inside it, then the airspace alone does not suffice; rather, the actual physical structure must be faced, such that even if he faces a projecting part of it measuring roughly two-thirds of a dhirā‘ (cubit) or more, this would be sufficient." [Ḥāshiyat ash-Sharwānī ‘alā Tuḥfat al-Muḥtāj, Vol.1/P.484]
Sheikh al-Qulyūbī (raḥimahullāh) said: "Unlike one who prays outside it — i.e., outside the Ka‘bah — for him the airspace (directly above it) suffices, even if he is higher than it, or even if he is at the location where it (hypothetically) has been demolished." [Ḥāshiyat al-Qulyūbī ‘alā Sharḥ al-Minhāj, Vol.1/P.154]
Accordingly, the prayer is valid on the upper floors of al-Masjid al-Ḥarām, even though these floors are higher than the roof of the Ka‘bah itself. And Allah, the Exalted, knows best.
I broke my oath and am now obligated to pay the expiation for a broken oath (kaffārat al-yamīn). Would it be valid to fulfill this by inviting the poor and needy to a meal for them to eat from?
Praise be to Allah, and peace and blessings upon our master, the Messenger of Allah.
The expiation for a broken oath (kaffārat al-yamīn) is given to the poor (fuqarā') and needy (masākīn), and it must be given in the form of ownership (tamlīk) transferred to them of the required amount — namely, one mudd of the staple food of the locality, or its equivalent monetary value. It isn`t valid to invite the poor or needy to a meal prepared for them to eat from, as this does not constitute "ownership" (tamlīk) but merely "permission to partake" (ibāḥah).
Imam al-Shirbīnī (may Allah have mercy on him) said: "What is intended is that ownership be transferred to them; therefore, providing them with lunch or dinner [as a meal to eat from] isn`t valid." [Summarized from Mughnī al-Muhtāj,Vol. 5/P.50]. And Allah the Almighty knows best.
Is it permissible to combine and shorten prayers at home before traveling?
It is impermissible to combine and shorten prayers at home before traveling because a person is called a traveler once he/she departs the borders of his/her country, and offering prayer that way was originally intended to exempt a traveler. And Allah Knows Best.