This definitely answers the specific question I was trying to help with: the driver is r8125. No doubt about it at all.
Sadly I have no answers for the next questions, ie. what is wrong with that driver or (as I suspect without much hard evidence) if your actual problem is at higher level.
This report seems to me to show that you have drivers (kernel modules) for two different ethernet interface devices. Do you really have two devices, one that uses the r8169 driver, and another that uses the r8125 driver? If not, I would fix that, but I’m really new at this stuff so maybe I just don’t understand the situation.