Indeed, that is a thing that should happen, especially since Sirius spent most of February convincing people he wasn't a bad guy.

Pretty much, yes. Mostly he was framed and has come back to London to find whoever did it, so that he can clear his name. My headcanon is fairly similar to canon in this regard, but left intentionally vague in the event that castmates who want to do something different show up. But he would like to not have to convince everyone that he meets that he's not a murderer.