Is it permissible for a financially capable woman to pay the Zakat to her sons or one of them?
It is permissible for a mother to give her children from zakat if they are eligible to receive it, even though she is obligated to provide for them. This is based on the saying of the Prophet Muhammad (PBUH) regarding Zainab, the wife of Abdullah Ibn Mas'ud (may Allah be pleased with them): "Your husband and your children are more deserving of your charity" [Al-Bukhari]. And Allah Knows Best.
Is it permissible for the doctor to uncover the hair of a female patient out of necessity?
It is only permissible for him to do so when there is no female doctor, and he should examine the patient in the presence of one of her Mahrams (unmarriageable kin).
Is permissible to buy/sell with non-Muslim?
Yes buying/selling for a Muslim with non-Muslim is permissible as regards what is lawful.
By slip of the tongue, my son cursed Allah. It is worth pointing that he has concluded his marriage contract recently, but haven`t consummated the marriage. What is the position of Sharia on this?
All perfect praise be to AllahThe Lord of The Worlds and may His Peace and Blessings be upon our Prophet Muhammad and upon all of his family and companions.
This person must return to the fold of Islam by uttering the two testimonies of faith and making repentance to Allah from all acts that lead to disbelief. Also, he must make a new marriage contract because the old one was dissolved after he had committed this act of disbelief. Moreover, since there is no waiting period for his wife, then she is in state of Talaq ba`en baynona Soghra (Minor irrevocable divorce); consequently, she can get back to him only by a new marriage contract. And Allah The Almighty Knows Best.