Shielded ethernet cable and ethernet sockets earthing how to?

Earthing is referred to as grounding in some countries. They are the same.

Shielding is protection from radio frequency (RF) interference, usually by a foil wrap around each pair or around the whole bunch of 8 wires (inside the sheath). Shielding works better when earthed (arguably it does not work properly when not).

Earth wire of the mains would be suitable assuming a) you can connect this safely, b) you only earth in one place. Grounding the shielded cable will NOT earth your PC at all, the earth wire of the power cable does that, or PC must be double insulated (eg if a laptop).

Use UTP

I would not suggest grounding the shield on the router end.

If for any reason the ground potential of a computer on the second floor is different than the ground potential of the router, then a ground loop will be formed via the shielding, possibly causing damage to either the cable or the router or the computer.