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.
Is wearing a cap permissible for boys, or girls?
All Perfect Praise is due to Allah, The Lord of The Worlds.
In principle, anything is permissible sol long as there is no Sharia evidence prohibiting it, thus wearing the cap by boys is permissible as long as it isn`t done in imitation of the disbelievers. Moreover, a Muslim woman should avoid all that which attracts the attention of men because whatever leads to the forbidden is forbidden. And Allah Knows Best.
A person is sponsoring an orphan and pays the sponsorship monthly. Is it permissible to pay this month's sponsorship from zakat money?
It is permissible to pay the orphan's sponsorship from zakat money if the orphan is poor and has no one to support them. The person must intend to count it as part of their zakat when paying the money. And Allah Knows Best.
An Arab young man had an illegitimate sexual relationship with a Christian American while she was staying in one of the Arab countries. After she returned to America, she told him that she had doubts that she was impregnated by him. It is worth pointing that it is hard for him to get a visa to the USA. What is the ruling of Sharia on this?
All perfect praise be to Allah the Lord of the Worlds. May His peace and blessings be upon our Prophet Mohammad and upon all his family and companions.
Zina (Adultery and extramarital relations) is one of the grave sins that incur the wrath of Allah if the adulterer didn`t make immediate repentance. However, the child isn`t attributed to the adulterer. Rather, he/she is attributed to the woman who got pregnant by him and, according to Sharia, it isn`t considered a legitimate child of the adulterer. And Allah The Almighty Knows Best.