What is the ruling on someone who prays Fajr late, after sunrise, knowing that they wake up on time but return to sleep and do not perform it within its time?
It is obligatory to perform the prayer within its prescribed time. If someone wakes up and knows how to act, they must perform it before the time ends. If they delay it, they are sinful and must make it up. And Allah Knows Best.
Is it permissible to swear by (Allah "The Living")?
Yes, it is permissible because "The Living" is one of Allah`s Attributes.
Is prayer mandatory on a dying person who goes through a state of unconsciousness, then wakes up?
If he wakes up and was conscious, then he should pray the way/manner he could, but if he wasn't, then he is exempted from obligatory prayer as Allah Says in this regard (What means): "On no soul doth God Place a burden greater than it can bear." [Al-Baqarah/286].
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.