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.
Is it permissible for one who sacrifices on behalf of another, with the latter's permission, to eat from the sacrifice?
It is permissible for someone who sacrificed on behalf of another with their permission to eat from it with their permission, and they stand in their place (act as their representative) in distributing it.
It is stated in Nihayat al-Muhtaj ila Sharh al-Minhaj (Vol.8/P.141): "And he—meaning the one sacrificing on his own behalf, provided he has not apostatized—has the right to eat from a voluntary sacrifice and his sacrificial gift; rather, it is recommended. As for an obligatory sacrifice, it is forbidden for him to eat from it, whether it was designated as such initially or as a liability in his dhimmah. And excluded by what has preceded is if he sacrifices on behalf of another, or if he apostatized, then it is not permissible for him to eat from it, just as it is absolutely impermissible to feed a disbeliever from it."(The text I found indicates the impermissibility of eating) ???
And it is stated in Hashiyat al-Jamal ‘ala Sharh al-Manhaj (Vol.5/P.262): "If a person offers a sacrifice on behalf of a living person with their permission, does he act as their representative in distributing it—on the grounds that permission to sacrifice is permission to distribute it—or does it depend on their explicit permission? This requires consideration, but the first view is not far-fetched." And Allah the Almighty knows best.
Is it permissible for the mother to give her children from the Zakah (obligatory charity) of her money?
It is permissible for the mother to give her children from the Zakah of her money if they are Zakah-eligible recipients, and she isn`t obliged to provide for them.
The imam recited a verse containing a prostration of recitation (*sajdat al-tilāwah*) but did not prostrate. After the imam's *salām*, a follower (*maʾmūm*) performed a prostration of forgetfulness (*sujūd al-sahw*) because the imam had not performed the prostration of recitation. Is it permissible to go against the imam in this way?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah.
There is no sin on the imam if he omits the prostration of recitation (sujūd al-tilāwah), even deliberately. The maʾmūm (the one following the imam) is obliged to follow the imam in the prayer, so if the imam does not perform the prostration of recitation, it is not permissible for the follower to prostrate, because the imam is appointed only to be followed.
If the follower introduces a prostration after the imam’s salām but before his own salām, in a case where there is no prostration of forgetfulness (sujūd al-sahw), his prayer is invalidated if he knew it to be prohibited. Otherwise, he is excused on account of his ignorance and his prayer is not invalidated.
It is stated in Ḥāshiyat al-Imām al-Jamal ʿalā Fatḥ al-Wahhāb: “His statement: ‘So the omission of the hayʾāt (the non-essential sunnahs of the prayer) is not compensated for by prostration; if one prostrates for any of them deliberately, his prayer is invalidated, unless he is excused due to his ignorance.’”
Since this issue is one that is hidden from the laymen, as it is among the subtleties of knowledge, and the laymen are excused for ignorance of whatever is of such a nature, and their act of obedience is not invalidated because of it, it may be said that the prayer of one who has not acquired knowledge of this issue is not invalidated, with the necessity of learning the rulings remaining. As for after knowing it, the prayer is invalidated. And Allah Almighty knows best.