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 required of one who doubts the number of rakʿāt during prayer?
All praise is due to Allah, and peace and blessings be upon our master the Messenger of Allah ﷺ.
If a person doubts the number of rakʿāt he has prayed, he should build upon the lesser number, as that is what he is certain of. He should then perform the prostration of forgetfulness (sujūd al-sahw) before the final salām at the end of the prayer. ʿAṭāʾ ibn Yasār narrated that the Messenger of Allah ﷺ said: "When any one of you is in doubt during his prayer and does not know whether he has prayed three rakʿāt or four, let him pray one more rakʿah and then perform two prostrations while seated before the salām. If the rakʿah he prayed was a fifth, these two prostrations will make it even; and if it was the fourth, then the two prostrations serve as a humiliation for the devil." (Reported by Abū Dāwūd.)
It is stated in al-Muqaddimah al-Ḥaḍramiyyah — one of the foundational texts of the Shāfiʿī school: "If one doubts whether he has prayed three rakʿāt or four, he is obliged to build upon the lesser number."
However, if such doubt recurs repeatedly and reaches the level of obsessive whispering (waswasah), he should not build upon the lesser number in that case — rather, he should build upon the greater number. And Allah the Almighty knows best.
What is the ruling of Islamic Law on eating or drinking during the circumambulation?
All praise is due to Allah, and peace and blessings be upon our master the Messenger of Allah ﷺ.
It is disliked (makrūh) to eat or drink during the circumambulation (ṭawāf).
Shaykh al-Islām Imām al-Nawawī states in al-Majmūʿ: "It is disliked to eat or drink during ṭawāf, and the dislikedness of drinking is lighter. Neither of them, nor both together, invalidates the ṭawāf. Al-Shāfiʿī said: 'There is no objection to drinking water during ṭawāf, and I do not consider it sinful; however, I prefer that it be avoided, as refraining from it is more befitting in terms of proper conduct.' Among those who explicitly stated the dislikedness of eating and drinking, and that drinking is the lesser of the two, is the author of al-Ḥāwī."
If, however, a person is in genuine need of drinking, then there is no dislikedness in doing so. In any case, the ṭawāf itself remains valid. And Allah the Almighty knows best.
Is a woman`s prayer considered invalid if non-Mahrams (marriageable men) saw her offering it?
A woman`s prayer isn`t invalidated if non-Mahrams saw her offering it, but she had better pray in isolation.