Is revealing some of the sins that the questioner became aware of on his cousin's and uncle's social media accounts considered backbiting (Gheebah) or exposure (Fadheehah), and is seeking forgiveness (Istighfar) and supplicating (Du'a) for them sufficient to be absolved from that?

There is no objection to accessing publicly available social media pages, and this is not considered spying, unless one knows that they contain something forbidden to view. However, informing the mother and siblings about a private act of another person is an act of exposing a Muslim, and the obligation was to conceal it. One can seek forgiveness for them and mention them with goodness instead of asking for their forgiveness, especially if the questioner is a girl who is a stranger to them.