What is the ruling on one who believes in something false out of heedlessness, such as believing in the predictions of an astrologer, and is that considered disbelief (kufr)?

Obsessive doubts when dealing with astrologers can cause great harm. If you believed what the astrologer said, repentance will erase your sin and make you as if you had no sin. You must advise your father and explain that astrologers do not know the unseen, and that knowledge of the unseen belongs to Allah alone, and call him to repent from believing them.