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 prayer mandatory on a dying person who goes through a state of unconsciousness, then wakes up?
If he wakes up and was conscious, then he should pray the way/manner he could, but if he wasn't, then he is exempted from obligatory prayer as Allah Says in this regard (What means): "On no soul doth God Place a burden greater than it can bear." [Al-Baqarah/286].
A man cursed Allah, is he allowed to perform prayer without making Ghusl?
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.
Cursing Allah is an act of apostasy and one who commits it must utter the testimonies of faith, make repentance, take back wife; however, it isn`t conditioned that he makes Ghusl (Full body ritual purification) to pray. And Allah The Almighty Knows Best.
Is the `Iddah (waiting period) of the divorced effective from the time the divorce took place, or from the time of registering it at the court?
It is considered effective from the very moment her divorce took place, but not from the time of registering it at the court. And Allah Knows Best.