Is it permissible to sleep after the time of the Fajr prayer has begun, with the intention of waking up shortly afterward?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah.
If a person sleeps after the time of Fajr has begun, with the second adhān, then sleeping is forbidden (ḥarām) at that point, unless he is certain that he will wake up and perform the prayer within its time. And Allah Almighty knows best.
What is the ruling on making up the regular pre-Dhuhr *sunnah* (*al-sunnah al-rātibah al-qabliyyah*) for one who missed it in the Dhuhr prayer?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah.
Every timed supererogatory prayer (nāfilah) is recommended (mandūb) to be made up if its time passes. As for a nāfilah that precedes the obligatory prayer, performing it before the farḍ is a matter of proper etiquette (adab); if it is prayed after the farḍ, it is still considered adāʾ (performed within its time). If the time of the farḍ has ended, it is recommended to make up (qaḍāʾ) the nāfilah for one who did not pray it.
Whoever misses the sunnah before Dhuhr, whether with an excuse or without one, it is recommended for him to pray it after the obligatory Dhuhr prayer, within the time of Dhuhr, even at its end, and it counts as adāʾ, not qaḍāʾ. It is also recommended for one who missed the pre-Dhuhr sunnah to pray it before the post-Dhuhr sunnah, in observance of the proper order. It is stated in ʿUmdat al-Sālik wa ʿUddat al-Nāsik (Vol.1/p.60): "What precedes the obligatory prayer has the same time as the obligatory prayer, and performing it before the farḍ is a matter of etiquette; after it, it is adāʾ. What follows the farḍ has its time begin with the performance of the farḍ and end with the end of its time."
If the time of Dhuhr has ended, it is recommended for him to pray both the pre- and post-Dhuhr sunnahs, and they count as qaḍāʾ for both. It is stated in ʿUmdat al-Sālik (1/61): "Every timed nafl, such as the ʿĪd prayer, Duha, Witr, and the regular rawātib of the obligatory prayers, if it is missed, it is recommended to make it up forever." And Allah Almighty knows best.
What is the ruling on a person in a state of major impurity (junub) walking or eating before performing the ritual bath?
It is from the Sunnah for a Muslim to hasten to perform the ritual bath for major impurity. However, if he delays it, he does not sin provided he does not miss the prayer. It is permissible for the junub to walk, eat, drink, and sleep. It is recommended for him to perform ablution before these actions to reduce the state of impurity, although the best is for him to perform the ritual bath. And Allah the Almighty knows best.
Is it permissible for the children of a deceased father to settle his debt from the Zakah (obligatory charity) money due on them?
It is impermissible to use the Zakah of one`s money for settling the debts of the deceased. However, children should settle the debts of their deceased parents out of filial piety. And Allah Knows Best.