Is it permissible to give lessons to mixed students of the seventh grade?
It is impermissible to give private lessons to mixed students since such an act leads to serious consequences for the students in this grade are teenagers.
I`m a university student. While I was sitting for an exam, the professor caught my fellow student cheating. I heard him telling her to hide the dossier. After the exam, he asked me to testify that I saw her cheating although I heard him telling her to hide the dossier but didn`t see her cheating. 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.
It was narrated that the Prophet (PBUH) said to a man: "Do you see the sun?" He replied: "Yes." He said: "Give witness in a similar case [which is as clear as the sun], or leave it." [Reported by Al-Baihaqhi in "Sho`ab Al-Eman" pp.10964]. Therefore, it is not permissible to testify about something that you haven`t actually seen. And Allah The Almighty Knows Best.
Is Iddah (waiting period) incumbent on an old lady whose husband had died?
Upon the death of the husband, Iddah is incumbent on the wife regardless of her age. And Allah Knows Best.
Is permissible to buy/sell with non-Muslim?
Yes buying/selling for a Muslim with non-Muslim is permissible as regards what is lawful.