explain how you are connected to the internet:
are you connected to a gateway (router/modem combination), or are you connected to a multi-part system (like a router that connects to a modem, or a switch that connects to a modem, etc.)?
what kind of connection are you on (DSL, Fiber, Cable, Satellite)? who's your ISP?
if DSL, how are you attenuated (VCI/VPI numbers -- this can usually be located in the advanced panel/WAN of your connection settings)? don't worry about it if you don't know; the rest of the information can help me guide you to figuring out more advanced info.
what are the makes/models of your router/gateway/however you're connected to the internet?
if you can find them, what firmware version(s) are you running?
are you wired or wireless?
what's the make/model of your NIC/wireless adapter?
what's your computer's operating system?
have you run any malware checks recently? if so, with what programs?
do you run any programs that modify your hosts file (Spybot: Search and Destroy, for instance)? in other words, do you run any "website immunization" programs?
do you run any "registry cleaners" or "game boosters"?
do you route your connection through a VPN or a proxy?
i'm going to need a lot more information than that to provide any sort of helpful advice, but these broad specifications will give me a basis to work with.
knowing that disabling your computer's firewall helped find servers, and knowing that you can't manually connect to any servers, one hunch i have is that you're running through a double NAT, but i need the above information before we can continue on that assumption.
a few really basic ideas before we go deeper:
run your IP through http://www.spamhaus.org/lookup/ and tell me the results
switch your DNS to a public DNS, like google's or OpenDNS.