If a demigod performs a great feat and the gods offer it to him/her, the demigod can be granted immortality, but it only happens once in a great while.
Yes. A good example of this is in the case of Jesus. His mother was a mortal and his father was a God. In order for ジーザス to become a God, he needed the blessing of a God and his mortal half to die. Only then, He could be reborn and become a God.