What is incumbent upon the one offering the sacrifice if, after slaughtering the animal, they discover that one of its internal organs is damaged or diseased?
All praise is due to Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
The presence of disease or defect in the internal organs of a sheep does not affect the validity of the sacrificial animal, unless the disease leads to the animal becoming emaciated and its meat becoming corrupted.
It is stated in al-Iqnā' (2/590) by Imam al-Shirbīnī: "The third disqualifying condition: an animal with a manifest illness — meaning one whose illness visibly results in emaciation and corruption of its meat. However, if the illness is minor and does not produce such effects, it doesn`t affect the validity of the sacrificed animal." And Allah Almighty knows best.
Is it valid for the 'aqīqah to be performed using the newborn child's own wealth?
All praise is due to Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
It is not permissible for the guardian to perform the 'aqīqah using the newborn child's own wealth, because the 'aqīqah is a voluntary act of giving (tabarru'), and a guardian is prohibited from making voluntary expenditures from the child's wealth. Should he do so, he becomes financially liable for what he spent. And Allah Almighty knows best.
What is the ruling on praying alone (munfaridan) in a row, and what should I do if I enter the mosque together with another person, and there is only a single gap in the first row with no second row present — should I fill that gap in the first row, or should I stand together with him in a new second row?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah ﷺ.
If a person enters [the congregation] and finds no space available in the row, it is recommended for him — after making his opening takbīr for the prayer — to gently pull someone from the row in front of him to pray alongside him, provided he senses that this person would be agreeable to it. If that person does not agree, then he prays alone, and there is no dislike attached to his doing so in that case.
If he ends up praying alone [behind the row], it is sunnah for him to stand directly behind the imam. Then, if someone else arrives, that person stands to his right; and whoever comes after them stands to his left — and so on — such that the imam remains centered in the middle of the row. This is done so that everyone attains the reward of congregational prayer.
As for the case where two people enter the prayer together, and there is a gap in the row wide enough for only one person, then the two of them should pray together in a new row [of their own], and whoever arrives after them should fill that original gap. This is because if only one of the two fills the gap, the other would end up praying alone [which is to be avoided when it can easily be prevented].
Al-Khaṭīb ash-Shirbīnī (raḥimahullāh) said in Mughnī al-Muḥtāj (Vol.1/P.494): "If he does not find space, then it is recommended for him, while still standing, to gently pull a single person from the row toward himself after making his opening takbīr — out of caution, so as to avoid the position held by some scholars that a person's prayer standing alone behind the row is not valid.
Az-Zarkashī and others said: this should only be done if it seems he would agree to it; otherwise, he should not pull anyone — indeed, doing so becomes impermissible, out of fear of causing discord (fitnah). It is recommended for the person being pulled to cooperate and agree, so that he may attain the reward of assisting in righteousness and piety (al-birr wat-taqwā). However, he should not pull one of two people [praying together] from the row, since this would cause one of them to end up praying alone. This is why the 'pulling' described here takes place only after the opening takbīr [i.e., once the person doing the pulling has already validly begun his own prayer]."
Accordingly: it is sunnah for you to pray alongside the person praying alone, and then, if someone else arrives afterward, you may go and fill the gap in the original row. And Allah, the Exalted, 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.