Is it permissible for one to shake hands with his uncle`s daughter?
No, it isn`t because he is a non-Mahram (Marriageable) to her.
Is it permissible to use the term 'Al-Jalalah' (Majesty) for anyone other than Allah, The Exalted?
In the Arabic language, a man of dignity and reverence is described as 'Jaleel' (noble), and similarly, an elder or a woman of dignity may be described as 'Jaleelah'. In such contexts, one may use the title 'His Majesty' or 'Her Majesty.' However, the Majesty of Allah, The Exalted, is unlike the majesty of humans, just as His hearing is not like our hearing, and His sight is not like our sight. And Allah The Almighty Knows Best.
Is it permissible to swear by (Allah "The Living")?
Yes, it is permissible because "The Living" is one of Allah`s Attributes.
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.