What is the ruling on believing and relaying to others news of slandering chaste women, and what are the sins and repentance associated with that, and what is the ruling on my relative who wrote it, and is it permissible to backbite a girl who openly commits فسق (transgression/immorality)?

This young man's informing you that the girl committed zina (fornication/adultery) is a grave evil and a form of slander (qadhf). Your believing him is also an evil. What is obligatory is to denounce him for this. The fact that the girl has bad manners does not mean she committed zina, and the default assumption is that she is innocent of it. You are sinful for believing this young man, and you must repent and frequently supplicate and seek forgiveness for this girl. It should be noted that this sin does not apply to you if it occurred before puberty.