It's not recorded in the Gospels but chances are they did since they were relatives.
We know from The Bible that Jesus was the only human that ever lived that was completely sinless. He did not need to be baptised. But His entire life was an example of how we should live. His baptsism was an example for us. We need baptism and 'rebirth' if we are to be saved.
Live in the desert, survive on bugs and honey, preach Jesus' coming and redemption through baptism. IE your not John the Baptist, nor will you or anyone else ever be.
Jesus and Cleopatra were not alive at the same time. Cleopatra died many years before Jesus was ever born.
No, baptism, if validly received, cannot be repeated according to most Christian denominations.
Jesus himself was baptized. I am sure he would have ensured his mother was also. However, I believe others can add to this. I just do not remember reading anything pertaining to Her baptism. Yes. Acts 1:13, 14. Acts 2 says they were all baptized in the name of Jesus.
Generally speaking, no one should ever be denied baptism in an Orthodox Church, as baptism is necessary for salvation.
No Jesus was never ever called Michael.
No. Jesus was ascended to heaven nearly 600 years before the advent of Islam. Islam was founded in around 610 AD; when the Quran was entrusted to Mohammad.
No. There is no record that he ever married.
There is no record of Jesus traveling to Rome.
No there was no need what so ever for Jesus to be confirmed.
No.