My guess is these two Ethernets don't connect to each other; they both go to some third location yet undiscovered. Back in the old days, there was no WiFi, and there was an Ethernet run to everywhere you wanted networking, the runs went to a single location where you had a switch, router, DSL modem, possibly a server etc. This equipment took at least a shelf or two and had a cluttered, cables-everywhere hacker aesthetic. Probably near a phone line outlet since DSL was still king. A homebrew version of the "wiring closet" you have in offices today.
I would look at locations in your house where that would make sense or be likely to be "wife approved". It won't be a place you would ever want to put a TV.
(Lots of) Home runs are good
You are correct that you want to run a cable to each room from the central switch. In fact, I would run at least 1 more cable than you think you will need to each room, and consider running a line or 2 to other rooms as well - especially if your walls are open. Cable is cheap, and pulling 4 cables instead of 3 is no more work when done at the same time. If you decide later that you want a 4th jack, you either need a small switch (which does limit bandwidth, not really increase latency) or you need to open walls again to pull that 4th cable.
Use a patch panel
Rather than run the cable from the big switch to each room, you should have a patch panel in between. Patch panels basically change the type of connection on the cable (the back is a 110 punch down block, front is an RJ-45 jack), and are a simple pass-through.
This is to ease installation. Pulling cable through walls is best done when the cable is un-terminated. Terminating the cable (i.e., putting the RJ-45 jacks on the end) can and is done, but punching the cable down into a patch panel is so much easier, especially for someone who has never done it before (and it sounds like neither you nor your electrician has). The cost is marginal (again, go bigger than you think you need now), but you save on headaches during installation.
You would then get keystone jacks that allow you to punch down the cable on the other end:
You shove these into wallplates on an electrical box or low-voltage plate:
They make wall-plates with different numbers of openings (usually 1-6), so you can get what you need for each room.
Finally, you would need short (1-2 ft) "patch" cables to connect the patch panel to the big switch. Buy these cables pre-made, as you won't be able to make your own for less. These are typically stranded cable, as it's more flexible.
Your final setup would look something like this:
(the top-most device with jacks is the patch panel, the middle on is the switch, and the bottom would be your router)
Buy solid copper UTP (unshielded twisted pair) cat5e or cat6 cable, rated properly (usually CMR for typical in-wall installation, but you'll need Plenum if you plan to run it in HVAC ducts), and buy multiple boxes if possible. Standard is 1000 ft but smaller lengths are available, and they come in all different colors. A decent-sized house could take 2000-3000 ft of cabling or more, depending on how many runs and where the network closet is. Again, the more boxes you have, the easier installation will be (you typically pull 1 from each box at the same time, so if you want 4 runs to a single location, having 4 boxes is easiest).
If you want things a little cleaner, you can get a wall-mounted mini rack as well:
Just make sure to get one that has the depth and vertical space (measured in "U") you need. They also make ones with hinges that make patch panel installation a bit easier.
Most product images taken from monoprice.com
Best Answer
There are digital stud finders that you use with your phone and the hype from their advertising implies you can see anything through anything. I can't vouch for them but it's worth a look.
There are wire tracers like the toneout boxes the phone company used to use. There are cheap ones. The receiver does not need electrical contact, it's meant to work through the wire insulation but I don't know how far through plaster it might work. Maybe worth a try.
Many VESA mounts are horizontally adjustable on their mounting screws so perhaps you could just drill outside the 10cm band and still get the result you want.