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.
What is the ruling on having brotherly ties between a strange man and a strange woman, and is the former considered a Mahram (unmarriageable) to the latter?
Such a relation between a strange man and a strange woman is forbidden in Islam because it involves forbidding what Allah has allowed by marriage, and allowing what Allah has forbidden such as looking and the like. Such a relation doesn`t render any act lawful between them, thus they are forbidden to look at each other, have a Khulwah (seclusion), and travel together.
Is it permissible for a sick person to pray while sitting on a chair?
All perfect praise be to Allah, The Lord of the Worlds It is permissible for a sick person, who is unable to stand up, to offer prayer while sitting on a chair. If he/she was unable to prostrate while on the chair, the he/she should pray while sitting on the ground, but if he/she was neither able to bow (Roku),nor to prostrate, then he can do them while sitting on the chair. And Allah Knows Best.