Next time you boot the PC go into BIOS and look under Integrated Peripherals and find Onboard LAN - Disable, save changes and continue boot.
You will not be connected to the network at all - look under Control Panel - Programs - for any drivers relating to the Onboard LAN card, remove them.
Reboot, re enable Onboard LAN and allow Windows to find and re install drivers, re connect to internet and see if there is any improvement.
Having read you post, carefully, I think the rest of the setup, laptop - router - outside world is fine, the problem does appear to be on the PC.
If you have a spare Network card, you could try that instead of Onboard LAN.