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.
What is the ruling on swearing a false oath by the Holy Quran?
Swearing a false oath by the Holy Quran dips the oath-taker in Hellfire, and one who had done so should turn to Allah in repentance, seek His forgiveness, give back rights to whom they belong, and pay the oath expiation.
Which hour coincides with the last third of the night?
This hour could be figured out by calculating the number of night hours which start at sunset, and end at dawn break, then the total is divided on three to get the result.
Is the marriage contract considered valid if concluded at home by the marriage official (Ma`zon)?
Yes, it is valid as long as it is registered at the court.