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 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.
Is it permissible for a person who hasn`t made ablution to call for prayer (Adhan)?
All perfect praise be to Allah, The Lord of The Worlds. It is disliked for a person with a minor impurity to call for prayer, but if he does , then it`s valid and sufficient. And Allah Knows Best.
How many prostrations of Quranic recital are there, and is it permissible not to offer them while reciting?
There is one prostration for the Quranic recital, and it is a Sunnah for which one is rewarded upon offering it. However, one who doesn`t isn`t punished. Therefore, those who fail to offer it aren`t considered sinful, rather they deprive themselves from the reward.