Well, what this tells us, is that the issue seems really limited to the browsers, not to the underlying OS, or something external (like a router gone bad).
I'd say double-check there isn't some proxy server configured:
Apple menu -> System preferences -> Network icon. From the list of devices click on Integrated Ethernet, then on the Proxies tab. No proxy should be configured.
Then if that's not the case, you could create another user and see whether it's also occurring there.
And another thing to check, would be to remove/install Firefox again, or download and install Google Chrome browser.