Is it permissible to swear by (Allah "The Living")?
Yes, it is permissible because "The Living" is one of Allah`s Attributes.
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 allowed for me to donate a flat to each of my children except one considering that the tuition fees I paid for the latter`s university study are equal to the price of the flat? In other words, he has received his share and this is why I`m donating a flat to my other children?
All perfect praise be to Allah the Lord of the Worlds. May Allah`s peace and blessings be upon our Prophet Mohammad and upon all his family and companions.
Yes, it is permissible because you want to treat them justly. And Allah The Almighty Knows Best.
Should one obey his mother even if she was wrong?
Obeying one`s mother is an obligation in matters that are beneficial to her and so long as she didn't drive her children to commit an act of sin.