Can a woman perform I‘tikaf in her home?
No, a woman's I‘tikaf is not valid if performed at home, because her home is not a mosque.
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 someone who dies while having missed fasts?
● If a person dies before having the opportunity to make up the missed fasts—such as someone whose excuse (e.g., illness) persisted until their death—then no makeup fast (qada), fidyah, or sin applies to them.
● However, if they had the ability to make up the fasts but did not do so before passing away, the missed fasts must be compensated by giving a mudd of food for each missed day from their estate.
The Prophet ﷺ said: "Whoever dies while having a month’s fast due, one needy person should be fed per day on their behalf." [Narrated by At-Tirmidhi]
Additionally, a guardian (wali) may fast on their behalf, as the Prophet ﷺ said: "Whoever dies while having missed fasts, their guardian should fast on their behalf." [Narrated by Al-Bukhari and Muslim]
In another narration: "If they wish." This indicates that both feeding the needy and fasting on behalf of the deceased are permissible options.
What is the ruling on performing dry ablution (tayammum) instead of the ritual bath due to severe cold?
Whoever is sick such that he cannot use water, or finds the water cold and heating it is not possible, and the person would be harmed by using cold water, it is permissible for him to perform tayammum. He must make up what he prayed with tayammum, because this is a place where there is water. And Allah the Almighty knows best.