Is it valid to perform the obligatory prayer while sitting?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
Standing (Al-Qiyam) while having the ability to do so is one of the pillars (Arkan) of the prayer; the obligatory (Faridah) prayer is not valid without it. However, whoever is unable to stand may pray sitting down. As for voluntary (Nafilah) prayers, it is permissible to pray them sitting even if one is able to stand, but the one sitting receives half the reward of the one who prays standing. And Allah the Exalted knows best.
What is the ruling on discharging one's zakāh obligation through the provision of a specific benefit (manfa‘ah) — such as performing a surgical operation for a patient, or providing housing to a tenant, and counting the value of this service as zakāh?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah ﷺ.
It is obligatory to pay zakāh from the actual asset (‘ayn al-māl) upon which zakāh is due, except in the case of zakāh on trade goods ('urūḍ at-tijārah), which is paid based on their monetary value. In all cases, zakāh must be given in the form of tamlīk (a full transfer of ownership), based on the words of Allah, the Mighty and Majestic (which mean): "Zakāh expenditures are only for the poor and for the needy, and for those employed to collect [zakāh], and for bringing hearts together [for Islam], and for freeing captives [or slaves], and for those in debt, and for the cause of Allah, and for the [stranded] traveler — an obligation [imposed] by Allah. And Allah is Knowing and Wise" [at-Tawbah/60]. The letter "lām" (لِ) in this verse — as in "lil-fuqarā'" ("for the poor") — indicates possession/ownership (lām al-milk).
It is not permissible to fulfill one's zakāh obligation by providing a benefit or service (manfa‘ah) — such as housing, education, or medical treatment — in place of the zakāh itself. This is because zakāh must be paid as wealth (māl) that is transferred into the ownership of the poor recipient, whereas such services are intangible benefits that are not, in themselves, capable of being formally transferred in ownership (ghayr qābilah lit-tamlīk). And Allah, the Exalted, knows best.
Is it permissible to divide one mudd of fidyah between two people?
No, it is not permissible to divide one mudd between two people, because it would not count as a full meal for either of them.
Does the 'aqīqah count as valid if it is slaughtered before the seventh day from the birth?
All praise is due to Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
The time during which it becomes permissible to slaughter the 'aqīqah begins from the moment the newborn is fully delivered from its mother's womb.
If the animal is slaughtered prior to the birth, it does not count as an 'aqīqah — it is simply considered an ordinary sheep slaughtered for its meat.
And Allah Almighty knows best.