FAO emcee47
Have been wrestling with Windoze since 3.1 and the one thing I know to be constant: you will never know it all.
If you can log into your router the problem becomes more interesting. Try to ping 127.0.0.1 as this will test your Ethernet card through the default loopback settings. Four replies means at least something is working. If you disable (right click the icon in Network Connections > 'Disable') or remove any second Ethernet card (not the onboard one); or have another machine, perhaps a wirelessly connected laptop, try to ping by computer name (I assume 'Home', from a previous post). This will show an address, if it is NOT within the range 192.168.1.x, then there is a clash between DHCP assignments.
You could try a hard reset of the router (a small inset button near the aerial, you may need to unfold a paperclip). Then log into it through your browser and reconfigure your internet access and DHCP settings, ensure that the range you set for assignment is all within the same subnet - if your router is 192.168.1.1, ensure the range is 192.168.1.x, would suggest you use 192.168.1.9 - 192.168.1.254 as this leaves 192.168.1.2 - 8 free for manual assignment if necessary.
If you can log into the router before a hard reset, check the DHCP assignment settings or, turn off DHCP and manually assign an IP address to your Ethernet card, again ensuring it falls within the correct range, as above.
Your default router login details: type '.routerlogin.net' after the usual first 10 characters of a URL in your address bar; the Username is: admin, Password: password (would suggest you change these).
Let me know the results and will see what else we can try.
With regard to the beeps and the long pause at boot, if your RAM passes the test, it's probably a graphics issue, I believe your monitor is connected to your nVidea GeForce2 MX card? The horizontal blue 15 pin connection on the back of your rig? If memory serves, this is an AGP card, it's worth checking the BIOS settings are set to use this card before the onboard VGA. You could remove the card from your motherboard and connect to the onboard VGA, see if that makes a difference. Before you try this, visit Gigabyte's website, find and install the latest onboard VGA drivers for your board. If you need, can provide brief instructions for safe removal of the card.
Let me know how it goes, eventually we'll crack it & you'll have learned something new.