Is it permissible for someone to make a supplication to Allah to marry an already married, or an engaged woman?
No, it isn`t, and the person who does so is sinful before Allah, The Almighty.
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 it permissible for a wife to leave her house without obtaining the permission of her husband, and not to come back unless he divorces her co-wife?
It is forbidden for the wife to leave without obtaining her husband`s permission, and it is impermissible for her to ask him to divorce her co-wife as this inflicts harm on the latter and such an act is forbidden in Islamic law.
Is the father a Mahram (Non-marriageable) to his son`s mother-in-law?
The father is a non-Mahram (Marriageable) to his son`s mother-in-law, so it is impermissible for them to look at each other, or to have a seclusion (Khalwah).