What is the official OO view about life after death? OO believe in purgatory?
OO does not believe in purgatory. After the death of the body the soul continues to live, waiting for the resurrection of the body and the final judgement which will occur at the second coming of Christ. The soul is neither in heaven or gehena at that time.
What OOs think about holy fire of Jerusalem? OOs believe that Armenian Patriarch can't get that fire?
The very awareness of Holy Fire outside of Jerusalem within the OO is limited. I have not seen any kind of formal statement about it.
In OO churches (especially Coptic Church), is permitted a layman baptize other person?
In extreme cases; that is when an unbatized person on his death bed, in the last few minutes of his life wants to be baptized and if a priest is not available; in the OO (Syriac tradition) a lay person can baptize another person in the name of the Father, Son and Holy Spirit.
The absolution in beginning of coptic divine liturgy, have same function of confession?
I cannot comment about the Coptic divine liturgy; I can comment about the Malankara Syriac tradition. The practice of absolution (Hoosoyo) at the beginning of the divine liturgy is a very new practice; that started in the 1980s. Holy Confession was mandatory before each time a person receives Holy Communion. So the number of people receiving Holy Communion by the late 70s was very low. Typically you had one or two people receiving Holy Communion. So in the 1980s, so as the encourage people to receive Holy Communion, this rigid requirement was relaxed and instead absolution (Hoosoyo) was granted. Now as a result the number of people receiving Holy Communion has gone through the roof; however the number of people Confession has gone down. Most people that I know confess only once or may be twice a year in Great Lent around Holy Week.. that is it.