OK here's what you have. The master socket is an NTE5. The back part which screws onto the backbox is the actual socket part, and it has a pair of wires going into it - this is your incoming line from outside. The bottom half, once the faceplate is off (the bit you had in your hand is the faceplate), contains the test socket. If all is wired correctly, with the faceplate off you should get a dial tone and broadband sync in the test socket, and all other extensions should be dead.
As you've seen, the extension wiring connects to the faceplate, which then plugs into the test socket. You clearly have an existing extension (the white/orange/blue wires connected to the back of the faceplate. This looks to be CW1308 twisted pair cable - which is the correct cable to use. Do you know where this goes to? Could you change your plans to put all your stuff on the end of this cable?
Here's what I'd do:
1) Remove the faceplate to expose the test socket
2) Plug a phone into the test socket and check for dial tone - you should get one
3) Plug the known working phone into all other extension sockets - they should be dead
4) Plug your router into the test socket (using a filter as normal).
You now have two options - depending on how comfortable you are accessing your router stats.
5a) Best option - connect to router web interface (192.168.0.1) in a web browser, click on 'Maintenance', log in with username admin and password sky and record the sync speeds. They should look something like:
Modem StatusConnected
DownStream Connection Speed 13107
UpStream Connection Speed 1105
5b) Less good option - do a speedtest on speedtest.net and record.
6) If you don't use the original extension, I'd remove this wiring from the faceplate. You'll probably then be able to get your new extension wiring into the faceplate.
7) Plug the faceplate back in.
8) Plug a phone into the new extension, and check for dial tone.
9) Plug the router in the extension and repeat step 5.
** I've just realised, you said fibre broadband at 20mb. You're already on a long line from the street cabinet (around 1km), so any additional cabling inside your house will seriously reduce this speed. If you do try the above, please only do it once, and make sure you disconnect the router from the line before you start making connections. Openreach systems continuously monitor the line, and will reduce the speed if instability is detected.
Is there any way you can keep the router at the master socket? This will allow you to get the VDSL filtered faceplate (mentioned before as interstitial faceplate), which keeps your broadband signal as clean as possible.
In fact, I imagine that just having that existing extension connected, even with the router in the master socket, will seriously impact your speed.
Cheers,
Jon