Is it permissible to offer prayer at home, or should it be offered in the mosque?
All perfect praise be to Allah, The Lord of The Worlds. It isn`t preferable for the man to pray at home as praying in the mosque is twenty seven times more rewarding. Therefore, this should motivate him to offer prayers in the mosque. And Allah Knows Best.
Is bleeding that lasts for nine or ten days considered menstruation (Hayd)?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
If the duration of the bleeding does not exceed fifteen days, then all of it is considered menstruation (Hayd), even if the regular cycle ('Adah) has changed. And Allah the Exalted knows best.
What is the ruling on reciting aloud (*jahr*) in a silent (*sirriyyah*) prayer?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah.
Reciting aloud (jahr) in the Fajr prayer, in the first two rakʿahs of Maghrib and ʿIshāʾ, and reciting silently (isrār) in the Dhuhr and ʿAṣr prayers is a sunnah. If the worshipper recites aloud where silence is prescribed, or the reverse, his prayer is valid, although it is better to follow the sunnah, in imitation of the Prophet (peace and blessings be upon him). Abū Hurayrah (may Allah be pleased with him) said: “In every prayer there is recitation. What the Messenger of Allah (peace and blessings be upon him) let us hear, we let you hear, and what he concealed from us, we conceal from you. If you add nothing to Umm al-Qurʾān (al-Fātiḥah), it suffices, and if you add to it, it is better.” Narrated by al-Bukhārī.
Shaykh al-Islām Imam al-Nawawī (may Allah have mercy on him) said in al-Majmūʿ (Vol.3/389): “The sunnah is to recite aloud in the two rakʿahs of Ṣubḥ (Fajr), Maghrib, and ʿIshāʾ, and in the Friday prayer, and to recite silently in Dhuhr, ʿAṣr, the third rakʿah of Maghrib, and the third and fourth of ʿIshāʾ. All of this is by the consensus of the Muslims, together with the mutually reinforcing authentic ḥadīths on the matter. This is the ruling for the imam. As for the individual worshipper (munfarid), it is sunnah for him to recite aloud according to us and according to the majority… If he recites aloud where silence is prescribed, or the reverse, his prayer is not invalidated, nor is there a prostration of forgetfulness (sujūd al-sahw) for it, but he has committed something disliked (makrūh).” And Allah Almighty 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.