Yes, you can see it online. That is the purpose of setting up the no-ip software. The software ensures that the domain name maps to the correct IP as you have a dynamic IP address. This domain name then allows you to access your systems remotely, whether it is for serving web pages, an FTP server, or even remote login access to your systems.
In order for the port forwarding to work, the port numbers have to match. If you want to run your webserver on port 9999, then you also have to forward requests to port 9999 (you will also have to request that you connect on port 9999 such as http://mysite.no-ip.org:9999). If you want to run your webserver on port 80, then modify your webserver configuration to accept connections on port 80.
Since you get the router's config page (or something similar) when you try to access the domain name, you most likely have an option for remote access to the router set to port 80. Make sure that you have the remote access options disabled. You may also have to change the port that the router listens on for access to the configuration page (if you change this port, you will also have to request that port in order to get to the configuration page: http://192.168.1.1:325).
I checked out the Abyss Web Server. It does look like a very easy to use piece of software. Thanks for mentioning …