Does Husband fasting while being impure for having sexual intercourse with his wife at night permissible?
Yes it is permissible for you to fast because impurity isn't among valid fasting conditions, but you are obliged to perform ritual impurity in order to catch up Fajr prayer at its due time. And Allah Knows Best.
Should one fulfill his vow of donating an animal to the poor and needy with one sheep although he had repeated the vow several times?
Each vow has to be fulfilled, and repeating the words of the same vow for confirmation renders its fulfillment obligatory as one vow. As for multiple vows, they have to be fulfilled as well.
Is it permissible to offer Salah (prayer) on behalf of the sick who is on his/her death bed, and is incapable of offering it?
It`s impermissible to offer Salah on behalf of the sick, or the dead.
I had my menstrual period at the beginning of Ramadan for six days, then it ended and I became pure. After four days, I started noticing some blood again, which has lasted for two days now, but it is not as heavy as menstrual blood. Is this blood considered menstrual blood, and what is the ruling regarding my prayers, fasting, and reading the Quran during this period?
All praise is due to Allah, and peace and blessings be upon our Prophet Muhammad.
The minimum duration of menstruation is one day and one night, and its maximum is fifteen days. Any blood beyond this period is considered irregular bleeding (Istihada). Since the bleeding did not exceed fifteen days, it is considered menstrual blood within the regular cycle. Therefore, you should not pray or fast until the bleeding stops and the signs of purity appear. If the bleeding stops before fifteen days from when it first started, then all the blood you saw is considered menstrual blood, and you must make up the fasts, but not the prayers. If it exceeds fifteen days, then the first six days are menstrual blood, and the blood that follows is considered irregular bleeding. Your prayers and fasting are valid, and there is no issue with them. And Allah Knows Best."
*This answer was updated on [18/5/2023].