Is permissible to buy/sell with non-Muslim?
Yes buying/selling for a Muslim with non-Muslim is permissible as regards what is lawful.
Is it permissible for a woman who is in her `Iddah (waiting period) due to the death of her husband to go out and mix with non-Mahrams (Marriageable men)?
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.
A woman in such a situation is only allowed to leave her house for a sound reason such as visiting the doctor, and she is only allowed to mix with her Mahrams (Non-Marriageable kin) since it is permissible for a woman, whether in her `Iddah, or not, to mix with her husband and Mahrams. However, she is allowed to accept condolences, and answer inquiries about her situation after such a calamity. And Allah Knows Best.
What is the ruling on determining the sex of the baby whether through killing the female spermatozoa, or any other way in order to have a male?
If it is done through having a certain type of food, taking a certain type of medication, or organizing the time of the marital relation, then it is permissible.
I said to my wife: "You are forbidden to me in bed as is my sister." 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.
You have to fast for two consecutive months before having an intercourse with her. If you can`t, then you have to feed sixty poor persons; a meal for each that is equal half a JD. And Allah The Almighty Knows Best.