Are children born after repeated verbal divorces between the parents, while they continued to live together, and then returned to each other with the permission of a sheikh, considered illegitimate children before God, and does this negatively affect their lives?

There is no need for worry or doubt regarding your lineage to your father. Your lineage to him is sound, based on the jurisprudential principle that certainty is not removed by doubt, and relying on the scholars' fatwa to your father that no divorce occurred. As long as your father cohabited with your mother believing the marriage to be valid and subsisting, Muslims are in agreement that if a husband engages in intercourse in a marriage he believes to be permissible, then the child born thereof is attributed to him. This is based on the Prophet's (peace be upon him) saying: "The child is for the bed, and for the fornicator is the stone." Your father should be gently advised regarding uttering divorce and anger. If there is a strong suspicion of magic, the treatment is through ruqyah shar'iyyah (Islamic exorcism).