To make a long story short, all my connection problems were caused by the Creative Labs X-Fi Extreme Music card I have, specifically the drivers for it.
If I enable the card (and thus drivers), all my problems return. Disable it, and they are gone!
I'm using my motherboard's on-board sound now without problems (nforce 3).
So if you want to test this, go to:
Start->Control Panel->System->Hardware Tab->Device Manager->Sound, video and game controllers
Find you creative card, right click on it and select Disable. Reboot (I had too), and try the game again and see if you have any connection problems.
Good Luck.
(see my other post for more information on the problems I was having)