Drywall – Cosmetic plug for 3/4″ hole in drywall

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.

What you have here is a "voice distribution panel", and it is just like a big dumb junction block for all the cables that are punched down onto it. What this does is allow you to plug in a phone to any of the jacks in the apartment, but it is NOT usable for data.

So, you can use those cables for data, but you will have to pull them all off that distribution block and terminate them with RJ-45 jacks or a patch panel and use short patch cables to connect to your switch. It looks like what I described is one of your questions written on a picture, so yes, you can do that.

Since it was wired for voice, the wall jacks may also need to be rewired. You'll have to open up the jacks you would like to use and see how they are terminated. My house was punched down to a T568 standard, but the apartment I was in only had 1 or 2 pairs connected like you would for 1 or two phone lines. Depending on how the jacks are wired, you could terminate the cables the same way in this box and not have to change them.

Keep the pictures that you took, and I would go ahead and punch everything back down the way you found it when you move out.