Is the statement that the brother of the young man with whom the wife was having a relationship said that her pictures were with his brother, considered a slander (qadhf), backbiting (namimah), or gossip (gheebah)?

The statement mentioned is not considered an accusation of adultery, but rather backbiting that necessitates repentance. There is no benefit in mentioning such words, and one must verify reported news. It is impermissible for the wife of the husband's brother to joke with him because he is a non-mahram to her, and it was your duty to advise her. Abandoning a Muslim brother is forbidden except for a legitimate reason, and reconciliation is religiously required, while compromise and forgiveness are essential. The better of the two is the one who initiates the greeting of peace. An apology from the one who erred is required, but it should not be a condition for reconciliation. Their failure to inquire about your well-being during hardship contradicts the requirement of empathy among Muslims, but do not hold it against them.