What is the ruling on performing the obligatory prayer immediately after the commencement of the Adhan?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
It is permissible to perform the obligatory prayer (Fard) immediately after the commencement of the prayer time. As the Adhan (call to prayer) serves as a sign that the time has entered, it is permissible to pray as soon as the Adhan begins. However, it is recommended (Mustahabb) for whoever hears the Mu’adhin to repeat what he says. If the Adhan occurs while one is already engaged in prayer, he should repeat the words of the Mu’adhin after he has completed his prayer. And Allah the Exalted knows best.
What is the ruling on applying perfume while fasting?
Applying perfume does not break the fast. However, it is better to avoid it, as fasting is a practice of simplicity and restraint, while perfume is a form of luxury.
Is it permissible to include others in the reward of one`s sacrifice?
In the Name of Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
It is permissible for the person offering an Udhiyah (sacrificial offering) to include others in the spiritual reward of the sacrifice. However, it is not permissible for multiple people to jointly share the purchase price of a single sheep or goat, unless one person gifts the money to the other.
It is stated in Mughni al-Muhtaj (6/137) by Al-Khatib al-Shirbini:
"If a person includes someone else in the reward of his sacrifice and slaughters it on his own behalf, it is permissible."
It should be carefully noted that including someone else in the reward of one's sacrifice is completely distinct from offering a sacrifice on behalf of someone else. And Allah the Almighty Knows Best.
Does using a respiratory device (oxygen) affect the validity of fasting?
Using an oxygen device does not affect the validity of fasting because oxygen is a gas with no physical substance.
However, if medicinal substances with physical particles are added to the oxygen, its use would break the fast, as it enters the body cavity through a natural passage.