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.
A man wishes to offer more than one uḍḥiyah (sacrificial offering), and intends by these sacrifices "on behalf of everyone whom Allah and His Messenger love." Is this intention valid?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah ﷺ.
The uḍḥiyah (sacrificial offering) is not valid with this intention, because if the sacrificial animal is from sheep or goats, it suffices for one person only; and if it is from cattle or camels, it suffices for seven persons.
However, one may sacrifice it on his own behalf, and then gift the equivalent of its reward to whomever he wishes.
Imam ar-Ramlī (raḥimahullāh) says: "Ash-Shāfi‘ī (raḍiyallāhu ‘anh) said: It is within the vastness of Allah's bounty, the Exalted, that He rewards the one giving charity as well. From this, the scholars of the madhhab said: it is recommended for a person to intend the charity on behalf of his parents, for example — for Allah, the Exalted, will reward them, without this diminishing his own reward in any way." He then relates, as the relied-upon position, the permissibility of saying: "O Allah, convey the reward of what I have recited specifically to so-and-so, and to the Muslims generally." Az-Zarkashī said: "For reward varies in degree — its highest form is what one designates specifically, and its lowest form is what one dedicates generally [alongside others]." [End of quote from Nihāyat al-Muḥtāj]. And Allah, the Exalted, knows best.
Is it permissible to purchase the Udhiyah on installments?
It is permissible to purchase the sacrificial animal (uḍḥiyyah) on installments or by borrowing its price. However, it is not recommended for the poor to do so, because they are not required to offer a sacrifice, and Allah does not burden a soul beyond its capacity. If doing so would lead to negligence in providing for one's dependents (nafaqah wājibah), then giving precedence to obligatory maintenance over borrowing to buy the sacrifice is appropriate. This is based on the saying of the Prophet (peace and blessings be upon him): "It is sufficient sin for a person to withhold food from those whom he is responsible to support."
Furthermore, the sacrifice is valid from one who has a debt, but it is preferable (awlā) to repay the debt first, especially if the debt is due immediately (ḥāll). And Allah Almighty knows best.
Is a person who forgot to make up missed fasts until the next Ramadan began required to pay expiatory compensation (Fidyah) alongside the makeup fast (Qada')?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah.
Whoever forgets to make up missed fasts of Ramadan (Qada') until the following Ramadan begins incurs no sin, pursuant to the statement of the Messenger of Allah (peace and blessings be upon him): "Indeed, Allah has pardoned for my nation their mistakes, their forgetfulness, and that which they are coerced into doing." [Narrated by Ibn Majah, and Al-Bayhaqi in Al-Sunan Al-Kubra]
Furthermore, such an individual is not obligated to pay a expiatory compensation (Fidyah). It is stated in Bushra Al-Karim (p. 580)—one of the authoritative Shafi'i jurisprudential texts: "As for delaying it [the makeup fasts] due to a valid excuse—such as travel, breastfeeding, forgetfulness, or ignorance of the prohibition of delay (even for one living among scholars)—no Fidyah is required. This is because delaying performance during the actual month is permissible due to these excuses; hence, delaying makeup fasts is even more permissible, even if it continues for years."
And Allah Almighty knows best.