Is it permissible for the person who hasn`t made wudu` (state of minor ritual impurity) to perform the Adhan?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
It is disliked (Makruh) for a person in a state of minor ritual impurity (Hadath Asghar) to perform the Adhan. However, if he does so, his Adhan is considered valid and fulfills the sunnah of the Adhan despite it being disliked. And Allah the Exalted knows best.
What is the Islamic ruling on the Udhiyah (sacrificial offfering)?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
The Udhiyah (sacrificial offering) is a Confirmed Sunnah (Sunnah Mu’akkadah) for every adult Muslim of sound mind who possesses the financial means, whether they are a resident, a traveler, or a pilgrim (Haj). This is based on the statement of the Prophet ﷺ: 'When the ten days [of Dhu al-Hijjah] begin and one of you desires to offer a sacrifice, let him not touch [cut] anything of his hair or skin' [Narrated by Muslim].
The point of evidence (Wajh al-Dalalah) here is that the Prophet ﷺ linked the sacrifice to the individual's will and desire by saying, 'and one of you desires.' This indicates that it is not obligatory (Wajib); had it been mandatory, he would have simply said, 'let him not touch his hair until he sacrifices' [without making it conditional upon desire].
Furthermore, it is narrated that Abu Bakr and Umar (may Allah be pleased with them both) would sometimes refrain from offering the sacrifice out of fear that people might mistakenly view it as an obligatory duty [Narrated by al-Bayhaqi and others with a good (Hasan) chain of transmission]. And Allah the Exalted knows best.
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 seeking forgiveness between the two Friday sermons?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
It is permissible to seek forgiveness (Istighfar) between the two Friday sermons (Khutbatayn), and it does not invalidate the Friday prayer (Salat al-Jumu'ah). Furthermore, it is not disliked (Makruh) to speak before the sermon, after it, or between the two sermons. Likewise, it is not disliked for one who enters during the sermon to speak if there is a need for it, provided he has not yet taken a place and settled into it. And Allah the Exalted knows best.