Is it permissible to slaughter a single sheep with the combined intention of both the uḍḥiyyah and the 'aqīqah?
All praise is due to Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
According to Shaykh al-Islām Imam Ibn Ḥajar al-Haytamī, it is not permissible to combine the intention of the uḍḥiyyah and the 'aqīqah in a single animal, as each of the two has a distinct and separate cause that differs from the other.
However, Shaykh al-Islām Imam al-Ramlī permitted the combining of both intentions in a single animal — and this position offers a degree of latitude and ease. And Allah Almighty knows best.
What is the ruling on mentioning Allah`s name upon slaughtering an animal?
Praise be to Allah, and peace and blessings be upon our master, the Messenger of Allah.
It is Sunnah for the slaughterer to say at the time of sacrificing the Udhiyah: (Bismillah - In the name of Allah); due to the saying of the Almighty: "So eat of that [meat] upon which the name of Allah has been mentioned" [Al-An'am/118]. It is not obligatory; if one leaves it out intentionally or forgetfully, the consumption of the meat remains permissible. And Allah the Almighty knows best.
What are the conditions for the validity of the slaughtering process according to Sharia?
Praise be to Allah, and peace and blessings be upon our master, the Messenger of Allah.
Regarding the Sharia-mandated conditions for the validity of the slaughtering process (Adh-Dhabh), they are as follows:
The Identity of the Slaughterer: The person performing the slaughter must be either a Muslim or from the People of the Book (Christian or Jewish).
The Required Cuts: Both the trachea (windpipe/breathing passage) and the esophagus (food passage) must be completely severed.
Stability of Life: The animal must possess stable life at the start of the slaughtering process. This is identified by clear signs, such as vigorous movement or the forceful gushing of blood after the throat and esophagus are cut.
The Tool of Slaughter: The tool used must be sharp, capable of cutting or piercing by its edge and not by its sheer weight or blunt force.
And Allah the Almighty 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. And Allah Almighty knows best.