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.
What is the ruling on forgiving a tenant next month's rent, with the intention of counting it toward zakāh al-māl?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah ﷺ.
It does not fulfill the obligation to count the rent owed on a house, or a portion of it, in lieu of zakāh. Rather, it is necessary to first transfer ownership of the zakāh amount to the poor person directly; only afterward may he be asked to pay the rent [from that money].
It is stated in ‘Umdat as-Sālik (p. 112): "If someone gives [money] to a poor person, stipulating that the recipient return it to him in payment of a debt owed to him, or says, 'I am making what you owe me into zakāh, so take it [as such]' — this is not valid. But if he gives it to him with the [unstated] intention that the recipient will use it to repay him, or says, 'Repay what you owe me, so that I may [then separately] give it to you as zakāh,' or the debtor himself says, 'Give it to me, so that I may [then] repay you with it' — this is valid, and there is no binding obligation for the recipient to actually fulfill this [informal understanding]."
We would also remind [the reader] of the ḥadīth of the Prophet ﷺ: "The Muslim is the brother of the Muslim: he does not wrong him, nor does he abandon him [to harm]. Whoever attends to the needs of his brother, Allah, the Mighty and Majestic, will attend to his needs. Whoever relieves a Muslim of a hardship, Allah, the Mighty and Majestic, will relieve him of one of the hardships of the Day of Resurrection. And whoever conceals [the faults of] a Muslim, Allah will conceal him on the Day of Resurrection." (Agreed upon). And Allah, the Exalted, knows best.
What is the ruling on speaking while relieving oneself?
It is disliked for one relieving himself to speak while doing so, except out of necessity. And Allah the Almighty knows best.
Is it permissible for someone who is unable to fast to pay the fidyah (ransom/compensation payment) to himself, due to his own poverty?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah ﷺ.
It is not permissible, nor does it fulfill the obligation, for a sick person who is currently and permanently unable to fast to pay the fidyah (ransom/compensation payment) to himself, since it is a financial obligation owed by him, established as a standing debt upon him — much like any other debt he owes. If he were to pay it to himself, it would not thereby be discharged. So if he cannot find [anyone eligible to] pay it to, it remains a standing obligation upon him, and is not thereby waived. If Allah, the Mighty and Majestic, later grants him wealth, it becomes obligatory upon him to pay it out to the poor and needy.
Imam al-Khaṭīb ash-Shirbīnī (raḥimahullāh) said: "There is no difference in the obligation of the fidyah between the wealthy and the poor person; its benefit lies in its remaining established as a standing debt upon the poor person [until he is able to pay it] — and this is the more correct position (al-aṣaḥḥ), as indicated by the wording of ar-Rawḍah and its source text." [Mughnī al-Muḥtāj, Vol.2/P.174]
Accordingly: it is not permissible for a Muslim to pay the fidyah to himself while he remains poor. If the sick person is poor, he is not obligated to pay the fidyah immediately, and it remains a standing debt upon him until a time when he has sufficient means and wealth. And Allah, the Exalted, knows best.