What’s the best way to try and troubleshoot the scenario when we have network connections to everything else but the Internet (i.e. external mail, MySQL, and FTP) works, on our IPCop run network? (And note in this case the IPCop run network usually runs perfectly fine, meaning we not delving into a setup that never worked before.)

Well, because you have all your other external networking protocols in operation, the first thing to verify is that your ADSL connection on the IPCop landing page does indeed show that you are connected (which it should be because you have access to everything else).

This then points to the proxy server giving issues and 90% of the time the solution is pretty simple:

Simply navigate to Services -> Advanced Proxy, scroll all the way to the bottom and hit the Clear Cache button to the right.

Working? Nifty.