First of all, clearing up origins, I think you'd wanna go with the "Dimensional Misplacement" idea, perhaps Yui's last act to get Shinji out of there. Don't go overboard explaining it, maybe have River pick up on Yui's gratitude as they're flying away from Unit 01's remains so she can have a clue about why Shinji's history and theirs don't match. Leave it at that if you can; no use murking the waters.
Now, to ideas... after Shinji wakes up and goes a little crazy... maybe he runs and screams, or maybe he thinks he's just hallucinating, or both... I can see River giving him a hug and telling him "Don't worry, it's okay now." completely out of the blue.
Jayne: Hey Doc, if your crazy sister is comforting somebody, what's that say about their sanity?
And of course, Mal and Jayne are going to be complaining about another crazy passenger. Though I think Shinji will calm down once he figures out he's not dead or crazy, and realizes he has a whole new lease on life. And of course, once he gets over moping about everyone he's ever known being dead.
River: Quit moping. You're giving me a headache watching you.
For the moment, I think he'll probably try to make himself useful. See if he can't make something palatable out of their food supply, perhaps. Clean up around the place, that sort of thing.
Jayne: What the hell happened to my room? How'm I supposed to find anything now!?
Hm... actually, I don't think he'd have that great of an effect on the crew, though the effect on Shinji would probably be drastic. It might be pretty funny the first time they start getting shot at, when Shinji realizes he's thrown in his lot with a bunch of criminals and that he's likely to be in the line of fire quite often.
Simon might enjoy having someone else who's "sane" around the ship. Kaylee and Inara might end up doing that mothering thing with him.