I will say that Little Sister sounds like a immature little twit. She's waiting to break up with her soon-to-be-ex-boyfriend (STBX) for reasons that are solely to do with her own convenience (she's worried about her finals, not his), but she still has time to shop for his replacement while she's continuing to live with him, letting him think that things are fine. And she's told her entire extended family and various friends about both the breakup and the replacement.
Seriously - if she's got time to start dating someone new, she's got time to pack up her stuff and move back home first.
In your place, if asked for advice I'd tell her that I felt really sorry for her STBX - she hasn't bothered to tell him she's breaking up with him, and she's already interviewing his replacement, and she's told everyone but him in her life about this. Other than that, I'd stay out of it, and I wouldn't contact Best Friend or STBX about the issue.
Cosmic justice would have soon-to-be-ex-boyfriends (STBX) finding out about the breakup from a mutual friend who asks him how he's doing since Little Sister dumped him. He comes home, they have a massive fight, and she flunks all her finals due to the fuss.
As far as the best friend thing goes - no, you can't ask her not to date him because it might change the friendship. If Best Friend knows that his new girlfriend is still living with her boyfriend and hasn't broken up with him yet, though, I'd say he's not all that great a catch either, and maybe they deserve each other.