Most of the coverage appears to've started after the indictment and other legal problems caught up with Libgen. Once they've started, waging a PR fight helps.
Not sure if there's a difference, but I've always thought that her site is SciHub, not Libgen?
Yes, she runs Sci-Hub. Sci-Hub eventually partnered with LibGen to help host files and reduce Sci-Hub's dependence on donated logins. (Previously all documents were accessed from their "real" journal/repository portals using credentials contributed by sympathetic academics.)