What is the semi-circular structure attached to the Holy Kabah?
The semi-circular structure attached to the Holy Kabah is Hijr Al-Kabah (is a low wall originally part of the Kabah) and passing behind it is an obligation while making Tawaf because it is a part of the Holy Kabah. And Allah Knows Best.
What is the ruling of Islamic Law on performing ṭawāf on the upper floors of al-Masjid al-Ḥarām, given that these floors are situated above the level of the Ka‘bah, and that the one performing ṭawāf passes by certain barriers, such as pillars and walls, during the ṭawāf?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah ﷺ.
There are numerous conditions for the validity of ṭawāf (circumambulation around the Ka‘bah), among them: that the one performing ṭawāf be inside the Masjid (al-Masjid al-Ḥarām) — even if the Masjid has been expanded, so long as the expansion does not extend out into al-ḥill (land outside the sanctuary boundaries). It suffices for the ṭawāf to be performed in the airspace of the Masjid or upon its roof, even if that roof is higher than the Ka‘bah (the House) itself.
Al-Khaṭīb ash-Shirbīnī (raḥimahullāh) said: "It [ṭawāf] is valid upon the roof of the Masjid, even if the roof of the Masjid is higher than the House — just as prayer [is valid] upon Mount Abū Qubays despite its elevation above the House. This is the relied-upon (mu‘tamad) position." [Mughnī al-Muḥtāj, Vol.2/P.246]
The validity of the tawaf isn`t affected by the pillars and walls standing between the one performing ṭawāf and the House, since the condition is only that the person performing ṭawāf be within the boundaries of the Masjid and outside the boundaries of the Ka‘bah itself.
Ash-Shirbīnī said: "It [ṭawāf] is valid inside the Masjid, even if it is expanded, and even if a barrier stands between the one circumambulating and the House — such as a water-supply structure (saqāyah) or pillars (sawārī)." [Mughnī al-Muḥtāj, Vol. 2/P.246]
Accordingly, ṭawāf performed on the upper floors is valid, and the presence of barriers such as pillars and walls during the ṭawāf does not affect its validity. And Allah, the Exalted, knows best.
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.
Is it permissible for someone who is unable to fast to pay the fidyah (ransom/compensation payment) to himself, due to his own poverty?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah ﷺ.
It is not permissible, nor does it fulfill the obligation, for a sick person who is currently and permanently unable to fast to pay the fidyah (ransom/compensation payment) to himself, since it is a financial obligation owed by him, established as a standing debt upon him — much like any other debt he owes. If he were to pay it to himself, it would not thereby be discharged. So if he cannot find [anyone eligible to] pay it to, it remains a standing obligation upon him, and is not thereby waived. If Allah, the Mighty and Majestic, later grants him wealth, it becomes obligatory upon him to pay it out to the poor and needy.
Imam al-Khaṭīb ash-Shirbīnī (raḥimahullāh) said: "There is no difference in the obligation of the fidyah between the wealthy and the poor person; its benefit lies in its remaining established as a standing debt upon the poor person [until he is able to pay it] — and this is the more correct position (al-aṣaḥḥ), as indicated by the wording of ar-Rawḍah and its source text." [Mughnī al-Muḥtāj, Vol.2/P.174]
Accordingly: it is not permissible for a Muslim to pay the fidyah to himself while he remains poor. If the sick person is poor, he is not obligated to pay the fidyah immediately, and it remains a standing debt upon him until a time when he has sufficient means and wealth. And Allah, the Exalted, knows best.