Use existing cat6 wiring in apartment for data
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.