Well I would not say by any means you need to put the server on its on computer, not just for this issue. I am sure there are plenty of people that host a server and play on the same one with no issues. Are you positive the computer is virus/rootkit free?
I would run combofix to run a scan, it should take less than 20 minutes. It would not hurt to run the free version of avast at least to check for a virus, remove it after if you need to. The Avast firewall may block eqemu server, but you could also turn off the shields for incoming traffic.
Honestly we really need more information, why not post a screen shot of your NAT on the firewall so we can look at how you have it setup, and post that config file anyway.
Last thing, you said it does this when you also play on the machine and then you said it does it when you surf the web. So if you do not do either one the server is fine?
And honestly I still think it is is router, do this search....
https://www.google.com/search?q=Netg...ient=firefox-a
If DD-WRT is available for that router you may want to try that. I also saw a post with a person saying their kindle fire hitting that router brings down all of their internet, I think it is the router.