Is it permissible to divide one mudd of fidyah between two people?
No, it is not permissible to divide one mudd between two people, because it would not count as a full meal for either of them.
What is the ruling on sacrificing a castrated animal or one with a missing tail?
Praise be to Allah, and peace and blessings be upon our master, the Messenger of Allah.
It is permissible to sacrifice a castrated sheep; as it has been established that the Prophet (peace be upon him) sacrificed:
"...two large, fat, horned, white-and-black, castrated rams (Mawju'ayn—meaning having crushed testicles)." [Narrated by Ibn Majah in his Sunan].
It is not permissible to sacrifice an animal that is missing its tail, rump, or udder due to being cut off. This is in contrast to an animal that was naturally born without a tail, rump, or udder; such an animal is valid for sacrifice. And Allah the Almighty knows best.
Does tooth extraction during the day in Ramadan break the fast?
Simply extracting a tooth during the day in Ramadan does not break the fast. However, if water or blood enters the body cavity, the fast becomes invalid.
Whoever's fast is invalidated in this way must refrain from eating and drinking for the rest of the day out of respect for the sacred month and make up for that day later.
It is preferable to postpone the extraction until nighttime or after Ramadan if possible.
How does one with a continuous condition (like urinary incontinence or non-menstrual vaginal bleeding - mustahada) perform ablution?
A person with a continuous condition must do three things:
1. Not perform ablution except after the prayer time has entered.
2. Change the bag or diaper placed to reduce the flow of urine or blood after the prayer time enters, wash the private part from blood or urine, and perform ablution immediately.
3. Perform the actions of ablution in immediate succession, then pray immediately without delay, unless he intends to pray with the congregation.
It is not permissible for a person with a continuous condition to combine two obligatory prayers with one ablution, as he must perform ablution for each obligatory prayer, even if making it up (qada'). And Allah the Almighty knows best.
