What is the ruling on offering an Udhiyah on behalf of another with their permission?
Praise be to Allah, and peace and blessings be upon our master, the Messenger of Allah.
It is permissible for a person to offer a sacrifice (Udhiyah) on behalf of someone else with their permission, even if the person performing it has not offered a sacrifice for themselves. It is stated in Sharh Manhaj al-Tullab (Vol.5/P.261) by Zakariya al-Ansari: "No one may offer a sacrifice on behalf of another without their permission... as opposed to when permission has been granted." And Allah the Almighty knows best.
Is the one offering the sacrifice liable if its meat spoils?
Praise be to Allah, and prayers and peace be upon our Master the Messenger of Allah.
If the meat (of the sacrifice) spoils due to the negligence of the one offering the sacrifice in preserving it, or due to improper storage, then he is liable to compensate for the portion due to the poor (which is estimated as half a kilogram of meat). If the sacrifice was a vowed (mandatory) one, then he is liable for all of it.
However, if he was not negligent, then there is no liability upon him, because its ruling is the ruling of a trust (like an item left in someone's care). And Allah Almighty knows best.
When does the time for the Fajr prayer begin?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
The time for the Fajr prayer begins once the Mu’adhin starts the second Adhan (Al-Adhan al-Thani), and not after the first Adhan. And Allah the Exalted knows best.
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.