It really doesn't matter what the runs to each room are for as long as they terminate in the wiring closet. Then, depending on which source you feed them, that's what will be in the room.According to what he said, there are two runs from the closet to each room. One run from each room is connected to the switch, along with one incoming (he never specified, but I would bet there are two incoming). He has said that the ISP offers an IPTV service, so my thinking is that the ports connected to the switch are intended for that. My ISP also offers this and the STBs grab IPv6 public IPs. They do not get cross-connected to the LAN in any way, because any connectivity for things like cloud DVR and "TV Anywhere" are handled by their servers and apps on your tablets, phones, computers, etc. There would then be a second set of cables for each room and a second incoming at the closet, and this is where the router would be connected. Then all of the "second" ports in all the rooms will be on the LAN. Keep it simple..
However, I did forget about the possiblity of IPTV being distributed via the switch vs Internet. But something tells me his system doesn't care and both can be on the same physical ethernet network (if he's even using it, which it doesn't sound like he is).
Putting the router in the closet will hurt wireless connectivity and removes the capability for more than one hardwired device in the living room. I don't think that's the solution the OP is going for.