Is an animal that results from crossbreeding between two types of livestock (an'ām) valid [for sacrifice]?
Praise be to Allah, and peace and blessings upon our master, the Messenger of Allah.
An animal that results from crossbreeding between two types of livestock (an'ām) is valid [for sacrifice], with consideration given to the higher of the two required ages. Thus, in the case of an animal crossbred between sheep (ḍa'n) and goats (ma'z), only one that has reached two years of age is valid [for sacrifice]. And Allah, the Most High, knows best.
Is the son of the sister by breastfeeding considered a mahram in Islam, and is it permissible for a woman to go for Hajj with him?
Everything that is prohibited by lineage is also prohibited by breastfeeding. The son of the sister by breastfeeding is considered a mahram, just like the son of the sister by blood, and it is permissible for him to be her mahram for Hajj and Umrah. 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 praying while wearing shoes?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
There is no sin for a Muslim to perform prayer while wearing their sandals or shoes, provided they are free from any impurities (najasa). It was narrated that Anas bin Malik (may Allah be pleased with him) was asked: 'Did the Prophet ﷺ pray in his sandals?' He replied: 'Yes.'" (Related by Al-Bukhari).
It is stated in Fath al-Bari (Vol.1/P.494) by Ibn Hajar (may Allah have mercy on him): "Regarding the phrase 'praying in his sandals,' Ibn Battal said: 'This is understood to apply as long as there is no impurity on them; furthermore, this is considered one of the legal concessions (rukhas)." And Allah the Almighty knows best.