Do you fall asleep with your protagonist?

Definitely! (It all comes out in revisions, of course, because you don't want to make the reader sleepy.)

I've heard that the "waking up in an empty, white room" sci-fi trope is about representing the blank page. So, anytime someone walks into an empty or blank space ...

If I don't watch it, I would start all scenes with somebody opening a door, or ringing a doorbell.
Not sure what the deeper meaning is there...