What is the ruling on a person in a state of major impurity (junub) or a menstruating woman (ha'id) reciting the Quran from memory?
It is not permissible for a menstruating woman, a postpartum woman, or a person in a state of major impurity to recite anything from the Quran, whether from memory, from the Quran, from a phone, or a computer. It is also not permissible for them to touch the Quran, based on what was reported from Ali bin Abi Talib that the Prophet (peace be upon him) was not prevented from anything regarding the Quran except major impurity (janabah). (Reported by al-Tirmidhi who said it is a hasan sahih hadith). Menstruation and postpartum bleeding are analogous to major impurity (janabah) as they are all major impurities (hadath akbar).
For those mentioned, it is permissible to mention Allah and supplicate even with verses from the Quran, provided they do not intend them as recitation of the Quran, but intend them as remembrance (dhikr) or supplication (du'a). And Allah the 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.
Is it permissible to combine and shorten prayers at home before traveling?
It is impermissible to combine and shorten prayers at home before traveling because a person is called a traveler once he/she departs the borders of his/her country, and offering prayer that way was originally intended to exempt a traveler. And Allah 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.
