Realtek 8111B Network Adapter Losing Connectivity only on main computer

[Shasoosh]

I own a P35-DS3 motherboard with the above LAN chip. Lately after an internet speed upgrade (to 50mps - don't know if that's related) I'm losing connectivity only in my main computer. I can still access the internet through wifi and I can still access the internet from other home computers connected by LAN but on my main computer ping will fail and I can't even access the router\modem interface with 10.0.0.138.
Right now my main suspect is the Realtek 8111B Network Adapter that seems to choke on high load. I got that behavior while downloading files from uTorrent and when copying from one computer to another over the LAN network.
When that happens the only thing that will "fix" it is a computer restart. If I'll take the router power cord out from the wall I'll get a BSOD, here's the interesting stuff the DMP file had to say:

Probably caused by : Rt630x64.sys ( Rt630x64+4e0e )
*********************************************************************

DRIVER_IRQL_NOT_LESS_OR_EQUAL (d1)
An attempt was made to access a pageable (or completely invalid) address at an
interrupt request level (IRQL) that is too high. This is usually
caused by drivers using improper addresses.
If kernel debugger is available get stack backtrace.
Arguments:
PROCESS_NAME: uTorrent.exe

My modem \ router is a Dlink 6740U-H1 with an updated firmware but I don't think that it's the cause of it.
Another thing I should mention is that sometimes when I get that behavior my Network name will change to unidentified Network.

Things I've tried:
Updating my Realtek driver to Win8 Auto Installation Program 8.012 2013/3/26 from here
Running those commands under cmd admin:

netsh interface tcp set global rss=disabled
netsh interface tcp set global autotuninglevel=disabled
netsh int ip set global taskoffload=disabled

Under my Realtek PCIe GBE properties ->Advanced
Speed & Duplex -> Mbps Full Duplex 100
Large Send Offload (iPv4) -> Disabled

Any help would be appreciated.