Are you certain that you have installed the r8168 driver successfully? Does it show up if you run
lsmod
Please see section beginning with “Realtek no link / WOL problem” at the link below. It may be of some help:
https://wiki.archlinux.org/title/Network_configuration/Ethernet
“Windows fast boot and BIOS: have read that this could also be a cause. Would rather not play with this as I am a noob.”
The above could be a lot of the problem here. Fast boot, hybrid sleep and Windows hibernation need to be disabled and Windows always needs to be shutdown completely in a dual boot setup. I would ask myself if you really do need dual boot or can you get by with Windows running in a virtual machine and avoid all of these headaches.
Some basic information here:
Please see section regarding Windows 10 preparation.