Does sacrificing one sheep avail for the entire household?
In the Name of Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
Offering an Udhiyah (sacrificial offering) is a communal Sunnah (Sunnah Kifayah) for members of the same household, provided that their financial maintenance is undertaken by a single provider. By "communal Sunnah," we mean that when one person performs it, the religious recommendation is fulfilled on behalf of the entire household, though the spiritual reward itself belongs uniquely to the one who offered it.
Therefore, if any member of the household performs the sacrifice—even if it is someone who is not legally responsible for the household's expenses, such as the wife or one of the children—the recommendation is fulfilled for everyone in that home. However, the reward does not automatically extend to the other members unless the person offering the sacrifice explicitly intends to share the reward with them—similar to how performing a funeral prayer (Janazah) fulfills the communal obligation for everyone, yet the specific reward is earned by those who actually prayed.
Additionally, a single sacrifice is sufficient for a man who is married to more than one wife. And Allah the Almighty Knows Best.
What is the correct point at which to raise the hands when standing up for the third rak‘ah (after the middle tashahhud)?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah ﷺ.
Raising the hands while saying the takbīr [when rising for the third rak‘ah] is a recommended sunnah, and omitting it does not affect the validity of the prayer. The proper time for this is after rising from the middle tashahhud, once one has fully stood up.
It is stated in I‘ānat aṭ-Ṭālibīn (Vol.1/P.158): "His statement, 'and rising from the first tashahhud' — meaning, this manner [of raising the hands] is also a sunnah when he rises from the first tashahhud, that is, when he stands upright from it.
Consider [the question]: when precisely does the raising of the hands begin? Is it at the very start of rising from the first tashahhud, or after he has reached the minimum position of rukū‘ [i.e., once fully upright]? The apparent, [correct] view is the latter — even though the outward wording [of the text] might suggest the former — because at the very start of his rising, he is still relying on both hands [to push himself up, and therefore cannot yet raise them]."
It is stated in ‘Umdat as-Sālik (p. 53): "If his prayer extends beyond two rak‘ahs, he sits after them in the iftirāsh position, recites the tashahhud, and sends blessings upon the Prophet ﷺ alone, without [mentioning] his family [at this point]. He then stands while saying the takbīr, supporting himself with his hands; and once he has stood upright, he raises them level with his shoulders." 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.
Should one who doesn`t perform prayer out of laziness make it up later, and how should he/she do so?
All perfect praise be to Allah,The Lord of The Worlds He/she is obliged to make up missed prayers by offering with each obligatory prayer another one, and if he/she offers two, then it is better. And Allah Knows Best.