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.
What is the ruling on congregational prayer performed outside the mosque, and is it a condition that it be performed in a mosque in order to attain the full reward?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah ﷺ.
It is not a condition for attaining the reward of congregational prayer that it be performed in a mosque; however, a Muslim should still be keen to perform his prayers in the mosque, so as not to miss out on immense reward. This is due to the virtue of the mosque and of spending time there (i‘tikāf), the typically larger number of worshippers present, and the additional reward gained through walking to and from the mosque. On the authority of Abū Hurayrah (raḍiyallāhu ‘anh), the Messenger of Allah ﷺ said: "Whoever purifies himself in his house, then walks to one of the houses of Allah to fulfill one of the obligations Allah has prescribed, one of his steps will erase a sin, and the other will raise him a degree [in status]." (Related by Muslim)
Whoever misses congregational prayer at the mosque should still strive to pray in congregation at his workplace, or together with the members of his household. And Allah, the Exalted, knows best.
Does buying meat equivalent to the weight of a slaughtered animal and distributing it avail for an udhiyah?
Praise be to Allah, and prayers and peace be upon our Master the Messenger of Allah.
That does not avail for the sacrifice (uḍḥiyyah), because the sacrifice necessarily requires slaughtering an animal from the category of anʿām (camels, cattle, and sheep/goats) within a specified time.
Rather, that is considered a form of charity (ṣadaqah) for which one is rewarded, but not an udhiyah. And Allah Almighty knows best.
Is it permissible for a woman to shake hands with her brother-in-law?
It is forbidden for the woman to shake hands with her brother-in-law because he isn`t a Mahram (unmarriageable) of hers.