When I use the "Test..." button to check for an external mail server, I receive a negative or error reply - what is wrong?

Could be different things, a firewall is likely one. But first of all confirm and reconfirm the mail server settings (hostname and / or IP-adress) and test again. Still negative? Check if the hostname of the mail server can be resolved - if not, use IP-adress only.

Test again, still negative? Then most likely the mail server is protected by a firewall which drops incoming ICMP commands. There is nothing you can do on the firewall settings, however: if all mail settings are okay e-mailing should still be possible behind a firewall.

If it is not working in your case, contact the administrator.