Is the woman who informs her husband about his sister-in-law's online conversations with young men committing a sin, is their discussion about the matter considered backbiting, and are they obligated to inform her husband?

As long as you have advised her and she has turned away, move on to another method, such as threatening to reveal her matter to her family. If she persists, then inform whoever has the authority among her relatives or guardians to prevent her from sin. If her husband is able to prevent her, there is no harm in informing him, and this is not considered prohibited backbiting or tale-bearing, because the intention is to change the evil. However, do not delve into the matter more than necessary, and do not spread the matter among people, because concealment is a great شرعي (Shar'i) objective. Informing your husband is not permissible because he has no authority over her, and informing him might lead to his recklessness.