Is permissible to buy/sell with non-Muslim?
Yes buying/selling for a Muslim with non-Muslim is permissible as regards what is lawful.
What husband's stand should be, if his wife disagrees with his mother?
Both the mother and the wife have rights on you (Questioner); therefore, you are obliged to treat both justly, resort to wisdom to please both and intiate reestablishing cordial relations among them. And Allah Knows Best.
Is Making up Missed Obligatory Prayers an Obligation?
All perfect praise be to Allah, The Lord of The Worlds, and may His peace and blessings be upon our Prophet Muhammad and upon all of his family and companions.
Missed obligatory prayer/prayers is a debt upon Muslim and making it up is an obligation, since the Prophet (PBUH) said: "pay the debt due to God, for it is the one which most deserves to be paid." [Agreed upon]. 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.