Does sacrificing one sheep avail for the entire household?
Praise be to Allah, and peace and blessings be upon our master, the Messenger of Allah.
In Islamic jurisprudence, the sacrifice (Udhiyah) is considered a Communal Sunnah (Sunnah Kifayah) for the members of a single household who share the same financial support.
If one member of the household performs the sacrifice—even if they are not the primary breadwinner, such as the wife or one of the children—the religious request is fulfilled on behalf of the entire household. This is similar to the Funeral Prayer (Salat al-Janazah), where the obligation is dropped for the community if some perform it.
While the communal request is satisfied by one person's action, the specific spiritual reward (Thawab) for the act of worship belongs only to the person who sacrificed, unless that individual explicitly intends to include the other family members in the reward.
A single sacrifice also avails for a man who is married to more than one wife. And Allah the Almighty knows best.
Is it a condition for I‘tikaf to be performed in the mosque?
Yes, for I‘tikaf to be valid, it must be performed in a mosque. I‘tikaf is not valid if performed at home or elsewhere.
Which is more excellent for a woman: to pray in the mosque or in her house?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
Our Master the Prophet ﷺ said: 'It is more excellent for a woman to pray in her house than in her courtyard, and more excellent for her to pray in her private chamber than in her house.' Therefore, a woman’s adherence to her home and her refraining from going to the mosques—in obedience to the command of Allah the Exalted—attains a great reward and abundant recompense. Furthermore, women praying in congregation within their homes is better than their attendance at the mosques, based on the aforementioned Hadith. And Allah the Exalted knows best.
What is the ruling on fasting the six days of Shawwal?
Fasting the six days of Shawwal is Sunnah. The Prophet ﷺ said: "Whoever fasts Ramadan and then follows it with six days of Shawwal, it is as if they have fasted for a lifetime." [Narrated by Muslim]
This is because fasting one month of Ramadan is rewarded as fasting for ten months, and the six days are equivalent to sixty days, completing a full year of fasting.