Is it permissible to offer prayer at home, or should it be offered in the mosque?
All perfect praise be to Allah, The Lord of The Worlds. It isn`t preferable for the man to pray at home as praying in the mosque is twenty seven times more rewarding. Therefore, this should motivate him to offer prayers in the mosque. And Allah 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?!
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.
What is the validity of the formula for sending blessings upon our master Muḥammad ﷺ: *"Allāhumma ṣalli wa sallim wa bārik ‘alā sayyidinā Muḥammad wa ‘alā ālihi, ‘adada kamāli-Llāhi wa kamā yalīqu bikamālihi"* ("O Allah, send blessings, peace, and grace upon our master Muḥammad and his family, by the number of Allah's perfection, and as befits His perfection")?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah ﷺ.
It is permissible to send blessings upon our master, the Prophet ﷺ, using the formula mentioned in the question, since sending blessings upon the Prophet ﷺ is an act of devotion legislated in its basic essence, without a specific prescribed wording required outside of the formal prayer itself. It is therefore valid to perform it outside of prayer using any formula. This particular formula involves multiplying the blessings sent upon the Chosen Prophet ﷺ to the number of the perfections of Allah, the Exalted — which are limitless — so that, by Allah's permission, its reward and benefit never cease.
The meaning of "the number of Allah's perfection" is that it is without limit, making the blessings sent boundless, corresponding to His perfection, Glorified is He, which cannot be enumerated or counted. This formula is analogous to the words of our Greatest Prophet ﷺ, as related in the authentic ḥadīth narrated by Imam Muslim: the Prophet ﷺ said to Sayyidah Juwayriyyah (raḍiyallāhu ‘anhā): "Indeed, I said four phrases after you [left], three times, which, if weighed against everything you have said since this morning, would outweigh them: Subḥān Allāhi wa biḥamdihi, ‘adada khalqihi, wa riḍā nafsihi, wa zinata ‘arshihi, wa midāda kalimātihi" (Glory be to Allah, and praise be to Him, by the number of His creation, by His pleasure, by the weight of His Throne, and by the extent of His words). And Allah, the Exalted, knows best.