What is the ruling on performing ablution (wudu') and ritual bathing (ghusl) with Zamzam water?
It is permissible to perform ablution and ritual bathing with Zamzam water. However, scholars disliked using it for cleaning after relieving oneself (istinja'). And Allah the Almighty knows best.
What do you say about someone who stays up all night on Laylat al-Qadr but misses Fajr prayer?
Such a person has deprived themselves of a great reward. Performing obligatory acts is more beloved to Allah than voluntary acts. Fajr prayer is an obligation, and performing it in congregation is equivalent to worshiping for the entire night. So how can someone neglect it while striving for voluntary prayers?!
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 multiplication of reward for prayer in the sanctuaries of Makkah and Madīnah limited to the obligatory prayer, or does it include the voluntary prayer as well?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah.
There is no difference between the obligatory (farḍ) and the voluntary (nafl) prayer in the ḥadīth on the multiplication of reward in the two Sanctuaries (al-Ḥaramayn), because the authentic ḥadīths are unrestricted and general in wording. It is therefore not permissible to restrict or specify them without evidence. Shaykh al-Islām Imam al-Nawawī (may Allah have mercy on him) said: “Our school holds that the superiority of prayer in the two mosques of Makkah and Madīnah is not limited to the obligatory prayer, but includes both the obligatory and the voluntary.” [al-Majmūʿ Sharḥ al-Muhadhdhab, Vol.8/p.476]
The multiplication of reward applies to whoever prays in al-Masjid al-Ḥarām or anywhere within the sanctuary (ḥaram). Al-Shirwānī (may Allah have mercy on him) said: “The more correct view according to al-Nawawī is that the multiplication of the prayer’s reward covers the whole sanctuary, and is not limited to the mosque, nor to Makkah itself.” [Tuḥfat al-Muḥtāj with the glosses of al-Shirwānī and al-ʿAbbādī, Vol.10/p.95]. And Allah Almighty knows best.