Ok, I had it for now with this r8169 kernel driver. I even tried a suggested newer kernel version but even that didn't help. I put in a new Intel card and it seems to work fine.
Because I switched ISP, I have to return my modem (not my property; Sagem BASE F@ST 3202). I still wanted to know if I could get access to superuser account again. It is still accessible by serial port for which there is no connector unfortunately. An phone data cable with some soldering helps in these cases.
Well.. I just cleaned up the pads I probed and screwing it back together.
Since I can't remember what cables it came with, I just put in the ones in worst condition :)
I also bought a WD MyBook 1TB disk.