In the room is the man,
This is grammatically correct, no doubt . "In the room" works as adverb clause and the verb and subject are inverted. In the normal way, it's written as
In the room, the man is.
There comes my question: how about
There's the room, in which is the man
In the sense that "In which" works as adverb clause and "the man is" is inverted. What do you think?
Unless there are rules which make is an exception, I think your sentence is fine, if a little odd-sounding.
For comparison, let's replace is with lies.
They all sound fine to me. But the fourth one sounds more idiomatic, especially if lies is replaced with the original is.
I think none of these expressions are in the usual, everyday word order, but all of them can be used in a poem, for example. It's about what you want to emphasize. Your third phrase is difficult to imagine on it's own; it would require something like "There's the room, in which is - THE MAN!" You would build up a suspense in the sentence, at the end of which you reveal the horrifying or surprising fact that THE MAN we have all been wondering about is in fact present. But if the sentence continues, it's not much of a strain to allow e.g. "There's the room, in which is the man who disappeared last year".
"in the room, the man is" clearly emphasizes the ROOM. He is in the ROOM, but not in the pool and the cellar and the attic all at the same time :)
