Running Coax and CAT6 through wall
Always hard to tell before opening things up, (and from limited pictures) but you appear to have a "thicker section of ceiling" out there in front of the closet, as compared to the underside of the presumed stairway (based on clues like the angled section of ceiling but it's a first floor location, and the open to second floor bit.)

If so, you can probably just punch one hole in the wall that goes through to that space, and one into that space from the closet, and have far fewer places to patch than if you run it through the studs, only one of which needs to be nice enough for company to see. Connecting between the holes is a matter of "fishing" a stiff-but-flexible fiberglass rod or spring-steel tape either through, or in from each side and hooking the two. I can't guess if that will be of any help with the room above, as I don't know the relationship of that room to this room.

That is, of course, if the space is not completely full of something else (like ductwork.) 

If you do go up to the attic, run a conduit to save trouble the next time - there's almost always a next time.