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Summarized Fatawaa

Is it permissible for a woman to offer the obligatory prayer after Adhaan (call for prayer), and before Iqamah (The announcement that the prayer is about to begin) in the mosque?
 

Yes, it is permissible for a woman to offer the obligatory prayer immediately after the Adhan and before the Iqamah in the mosque. And Allah Knows Best.

Does the intention (niyyah) of the principal (muwakkil) suffice for the sacrifice (dhabḥ) of the udhiyyah?

Praise be to Allah, and peace and blessings upon our master, the Messenger of Allah.
If a person appoints an agent (wakīl) to perform the slaughter (dhabḥ) on their behalf, the intention (niyyah) of the principal (the one appointing the agent) suffices, and there is no need for the agent to have this intention himself. In fact, even if the agent is unaware that he is performing the sacrifice on someone's behalf as an udhiyyah (sacrificial offering), this does not affect its validity. And Allah, the Most High, knows best.

What are the categories of the livestock permissible for Udhiyah?

In the Name of Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
 
An Udhiyah (sacrificial offering) is only valid if it is selected from Al-An'am (livestock), which includes: camels, cattle, and sheep or goats—whether male or female, and even if castrated. Allah Almighty says {what means}: "And for all religion We have appointed a rite [of sacrifice] that they may mention the name of Allah over what He has provided for them of [sacrificial] livestock." (Al-Hajj/ 34)
 
In terms of preference, camels are considered the best choice, followed by cattle, then sheep, and finally goats.
A single sheep or goat is valid as a sacrifice on behalf of one person. However, a single camel or cow can be shared by up to seven people. This is based on the narration of Jabir (may Allah be pleased with him), who said: "In the year of Al-Hudaybiyah, we sacrificed along with the Messenger of Allah (peace and blessings be upon him) a camel on behalf of seven people, and a cow on behalf of seven people." (Narrated by Muslim)
 
(Note: The term "Badanah" used in the original Arabic text specifically refers to a camel in this context). And Allah the Almighty Knows Best.

What is the ruling on performing dry ablution (tayammum) instead of the ritual bath due to severe cold?

Whoever is sick such that he cannot use water, or finds the water cold and heating it is not possible, and the person would be harmed by using cold water, it is permissible for him to perform tayammum. He must make up what he prayed with tayammum, because this is a place where there is water. And Allah the Almighty knows best.