Thanks @danielp for the very thorough and thoughtful answer!
In general I'm following what you're saying, however I have one followup question if you don't mind. And, it probably stems from me just not being familiar enough (yet) with the jME3 API and its underlying concepts.
At first you answer my initial question ("I'm wondering which would be applying the transformation updates/game actions: the client or the server.") as follows:
"The simple answer is... yes. Both of them do."
But then you answer my last question ("It is my understanding that the server does not run/manage its own independent scenegraph.") as follows:
"In general, yes. As @thetoucher said, a scenegraph is usually used for displaying the visual representation of the objects in your game so it doesn't make much sense to have one on the server."
To me (a poor, hapless newb!) these statements feel/seem contradictory, although I'm sure you're referring to 2 different things in each statement.
So I guess I'm tied up with not understanding the difference between the "application of transformation updates" and "scenegraph management". It seems that the former typically occurs on both client and server, but that the latter is almost always (your ray tracing example excluded) conducted client-side. If that's true, then I guess the root of my question here is: why?!
How can the server apply transformation updates if its not also running its own server-side scenegraph?
Thanks again so much for all the help so far!