When is it Sunnah to slaughter the 'aqīqah?
All praise is due to Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
It is Sunnah for the 'aqīqah to be slaughtered on the seventh day from the birth of the newborn. According to the sounder position, the day of birth itself is counted as the first of the seven days. Thus, for example, if the child is born on a Saturday, the 'aqīqah is to be slaughtered on the following Friday. If the child is born at night, the count begins from the day that follows. And Allah Almighty 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 are the Sunnahs and etiquettes related to the slaughtering of an animal?
Praise be to Allah, and peace and blessings be upon our master, the Messenger of Allah.
1-The animal should not be terrified or frightened before it is slaughtered.
2-An animal should not be slaughtered in front of other animals.
3-It is recommended to offer water to the animal before it is slaughtered.
4-The knife must be sharpened before the slaughter to ensure that the esophagus, trachea, and jugular veins are severed as quickly and cleanly as possible. This is based on the Prophet’s (peace be upon him) saying: "Verily, Allah has prescribed Ihsan (excellence/proficiency) in all things. So, if you kill, kill well; and if you slaughter, slaughter well. Let each one of you sharpen his blade and let him spare the animal unnecessary suffering." [Narrated by Muslim].
5-One must not begin skinning or butchering the animal until its death has been fully confirmed.
6-Cows and Sheep: It is best to slaughter them while they are lying on their left side, facing the Qiblah. The right hind leg should be left untied so the animal can move it, which helps it remain more comfortable during its final moments. Unlike other livestock, it is preferred to perform Nahr (slaughtering by a swift stab at the base of the neck) while the camel is standing, with its left knee tied. And Allah the Almighty knows best.
Is it permissible to participate in an Udhiyah?
Praise be to Allah, and peace and blessings be upon our master, the Messenger of Allah.
Regarding the sacrificial offering (Udhiyah), if it is a sheep or a goat, it only avails for one person. However, if it is a cow or a camel, it avails for seven people. It is permissible for the seven participants to have different intentions; for example, some may intend the sacrifice (Udhiyah), some the ’Aqiqah (sacrifice for a newborn), and others may simply intend to obtain meat.
It is stated in Hashiyat Qalyubi ‘ala Sharh al-Mahalli (Vol.4/P.256) and in Bushra al-Karim (p. 706):
"If one slaughters a camel or a cow on behalf of seven... it is permissible. Likewise, it is permissible if some of them intend their seventh for meat, and others intend their seventh for ’Aqiqah." And Allah the Almighty knows best.