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.
Is prayer nullified by nose bleed?
All praise be to Allah, The Lord of The Worlds. Blood coming out of the nose, or a wound doesn`t nullify prayer regardless of its quantity. And Allah Knows Best.
Is the father a Mahram (Non-marriageable) to his son`s mother-in-law?
The father is a non-Mahram (Marriageable) to his son`s mother-in-law, so it is impermissible for them to look at each other, or to have a seclusion (Khalwah).
What is the ruling on using toothpaste during the day in Ramadan?
Using toothpaste does not break the fast as long as nothing reaches the body cavity. However, it is disliked (makruh).
It is preferable for a fasting person to use it before the time of fast starts or after Iftar (breaking the fast) to avoid risking their fast.