I don't know what sort of services no-ip.com provides. If they provide a dns service to their customers then that should work... however you need some sort of configuration tool (web based?) to setup your domain information on their DNS servers.
Basically before you even begin thinking about a mail server, make the DNS work first. Its impossible to test the mail server effectively without it.
Try this to test your mailserver dns settings.
$ dig mx yourdomain.com
;; QUESTION SECTION:
;yourdomain.com. IN MX
;; ANSWER SECTION:
yourdomain.com. 10800 IN MX 10 mx1.yourdomain.com.
;; AUTHORITY SECTION:
yourdomain.com. 10791 IN NS ns1.yourdomain.com.
;; ADDITIONAL SECTION:
mx1.yourdomain.com. 10800 IN A 188.8.131.52
If you don't get all three of these in the response then you're not setup with DNS properly. The IP address 184.108.40.206 above should be your IP address.