My timezone is +10. Yes, the e alias just makes it easier to "e filename". I also use a "se" alias that just prepends a sudo to easily edit root owned files.
Trying to add temporary domainname mapping to your desktop and server is not really necessary. You should try to solve the basic networking problem of being able to "see" your DMZ server by going to your external IP from your desktop machine.
Run the tcpdump command on your server and then try to surf to your external router IP on your desktop and see what tcpdump says, or not. It should be obvious whether you see any sign of network traffic, coming from your desktop, on your server. We just need a yes or no answer.