Is vomit among the nullifiers of Wudu (ablution)?
Vomit does not nullify Wudu, but it is a Najaasah (impurity) that requires rinsing the mouth and washing whatever became dirty by it since the prayer becomes valid only after the removal of Najaasah. And Allah Knows Best.
A pregnant woman has iron deficiency. When she showers, she becomes very tired and feels cold. What is the ruling on tayammum in this situation?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah ﷺ.
One should not resort to tayammum (dry ablution) as a substitute for the obligatory ghusl (ritual bathing) so long as it remains possible to use water — for example, by heating the water, warming the place where the bathing takes place, or taking other similar precautions. However, if such measures prove ineffective, or are simply not possible to carry out, and performing ghusl [with cold water] would result in severe harm, then tayammum becomes permissible in that case, due to genuine inability to use water on account of the harm that would result, as described in the question. That said, she must make up (qaḍā') the prayers she performed using this tayammum.
It is stated in Bushrā al-Karīm: "One should not perform tayammum on account of cold unless warming the limbs fails to remove the aforementioned harm, and no means are available to heat the water — such as a vessel, firewood, or fire — and one fears [that using cold water would result in] any of the harms mentioned earlier in the discussion on illness: danger to life, loss of function in a limb, the onset or worsening of an illness, delayed recovery, or the [visible bodily] blemish mentioned in that discussion on illness. In such a case, tayammum becomes permissible due to the harm involved, but the prayer must still be made up afterward (qaḍā'). If, however, warming the body proves effective, or the means to heat the water are available, or none of the aforementioned harms are feared — then tayammum is not permissible, even if using the water causes some degree of discomfort or pain, since mere discomfort alone does not permit resorting to tayammum." And Allah, the Exalted, 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 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.