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 a condition that one should offer two sacrifices for a male newborn?
It is desirable to offer two sheep for a male newborn, and one for a female newborn. Offering one sheep for the male newborn is also permissible as it observes the Prophet`s Sunnah in this regard.
What is the ruling of Islamic Law on applying perfume before entering the state of ihram
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah.
Applying some perfume (ṭīb) before entering the state of iḥrām does not affect the validity of the Ihram, even if its trace remains after the iḥrām has begun. Indeed, it is sunnah for the pilgrim performing Ḥajj or ʿUmrah to perfume himself before iḥrām.
The scholar Nawawī al-Jāwī (may Allah have mercy on him) said: “The third of them is applying perfume to the body shortly before it, that is, before iḥrām, whether the one entering iḥrām is male or otherwise, young or old, married or unmarried. The recommendation applies after performing the ritual bath (ghusl), and it is fulfilled with any perfume whatsoever, though the best is musk, mixed with rose water or the like. There is no objection to the perfume remaining on the body after iḥrām; however, if a woman becomes bound by the mourning observance (iḥdād) after entering iḥrām, she must remove it. As for perfuming the garment, it is merely permissible, not recommended, according to the reliable position (al-muʿtamad). But if one removes his perfumed garment while the scent is still present in it and then wears it again, he is liable for the fidyah (expiation).” [Nihāyat al-Zayn, Vol.1/212]. And Allah Almighty knows best.
Who is the one required to slaughter the 'aqīqah?
All praise is due to Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
It is Sunnah for the 'aqīqah to be slaughtered by the guardian of the newborn — the one upon whom the child's financial maintenance is obligatory — provided he is financially capable of doing so. 'Alī ibn Abī Ṭālib, may Allah be pleased with him, reported: "The Messenger of Allah ﷺ performed the 'aqīqah for al-Ḥasan with one sheep." — Narrated by al-Tirmidhī. And Allah Almighty knows best.