Thanks for this tip, I'll try ASAP.

I followed your instructions but now, by default when I restart eth0, I'm in 10mb half duplex. I can set it manually to 100mb full. I still can't set it to 1gb.

Well, at least you tried an alternative. You might want to revert the changes and use the kernel provided r8169 instead.
There are other non OS/driver related things you might want to try:
- enter the BIOS settings and disable Wak-on-LAN & Network booting if they're active, maybe you have also some Ethernet related settings
- check if you have a BIOS (firmware) update and go for it
- disable auto-negotiation and enforce Gigabit (if possible) on the device you connect your host to (the other end of your Ethernet cable)
- wait for new kernel (r8169 driver)