How is night prayer offered?
Night prayer is offering voluntary prayer after Maghrib and before Fajr (Dawn). As for Tahajjud, it is offering voluntary prayer at night after waking up voluntarily, and for Allah`s sake as He Says (What means): "And pray in the small watches of the morning: (it would be) an additional prayer (or spiritual profit) for thee: soon will thy Lord raise thee to a Station of Praise and Glory!" [Al-Isra`/79]. Offering Tahajjud is better than offering voluntary prayer before going to bed.
What is the ruling on offering the sacrifice (uḍḥiyah) using the wealth of a minor or a legally incompetent person (safīh)?
Praise be to Allah, and peace and blessings upon our master, the Messenger of Allah.
It is not permissible for the guardian (walī) of an orphan or a legally incompetent person (safīh) to offer the sacrifice (uḍḥiyah) on behalf of the minor or the safīh using their wealth, because the guardian is charged with exercising caution over their wealth and is prohibited from giving any of it away voluntarily, and the udḥiyah is a voluntary act of charity (tabarru'). The guardian may, however, offer the sacrifice using his own personal wealth. And Allah, the Most High, knows best.
Is it correct that everything dry is pure even if it has impurity on it?
If something impure becomes dry, it remains impure and is not purified by drying. However, the impurity does not transfer by touching it if the one touching it is also dry. And Allah the Almighty knows best.
When is the prayer considered *adāʾ* (performed within its time): upon rising from the bowing (*rukūʿ*), or after rising from the prostration (*sujūd*)?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah.
Whoever catches one complete rakʿah within the prayer's time has caught the prayer as an on-time (adāʾ) prayer. The rakʿah is complete when the worshipper raises his head from the second prostration (sajdah), even if he has not yet reached the degree of standing in which the recitation is valid.
It is stated in Nihāyat al-Muḥtāj (Vol.1/p.378): "What is meant by the rakʿah is the completion of all of it, with its two prostrations." End quote.
And in Ḥāshiyat al-Shabrāmallisī: "(His saying: 'a rakʿah') means that he raises his head from the second prostration, even if he has not reached the degree in which the recitation is valid." And Allah Almighty knows best.
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Note on terminology: Adāʾ refers to performing the prayer within its prescribed time, as opposed to qaḍāʾ (making it up after its time has ended).