Cannot open the site at the specific port

Perhaps you have not provided enough info to fully answer this. Here goes a try.

You have a web server on a home machine configured to accept inbound connections on port 8080. Let's assume its IP address locally is 192.168.1.123. You have port 8080 forwarded in your router to that machine at the .123 address. You are trying to connect to your website from the external side of the router, over the Internet.

My first test would be to use a port scanner to see if 8080 is truly open and accepting connections from the Internet. One such tool is at Steve Gibson's site http://www.grc.com . From there choose Services and then ShieldsUP! And from there test port 8080.

If 8080 is truly open and accepting connections then to test where this is failing I would try the following from the 192.168.1.123 machine:

From your preferred web browser:

-- http://192.168.1.123:8080
-- http://foo.my-hobbysite.com:8080 OR -- http://external_IP_address:8080 (external IP address can be found from the DynDNS Updater screen under Current IP address)

The reason port you are getting a router configuration page when you try port 80 is likely that you have configured your router to accept remote mainentance requests on that port. I'd be changing it to something a little more difficult to guess and would be ensuring I have a secure password on it. If you don't plan to use that feature -- TURN IT OFF.

Hope this helps.

If you use just "http://foo.my-hobbysite.com/", your browser uses port 80 for HTTP as default, and this is where your router is listening. (You have to secure this, i.e. see if you can configure the router to not present its webinterface then. This is obviously a security risk. If you can't avoid it, you have to secure the router with a very strong password.)
Therefore you must use the format "http://foo.my-hobbysite.com:8080/", either directly or via a WebHop redirection.

As you already tried, you can use the Open Port Tool to see if your port 8080 can be reached and your application is listening on it. You should try with both, your DynDNS hostname and your WAN IP address, and always with port 8080, of course.
Before doing this and anything else with your DynDNS hostname, you must check that it is updated with your current WAN IP address.
If the result from the Open Port Tool or such is positive, then the problem is most likely with your application not accepting a connection with at least one of your DynDNS hostname or WAN IP address. Look into the documentation of your application to see how this is to be configured.

Also, it is a good idea to work through this wiki.

"Also, my local firewall has been turned off."
This is not a good idea and a big security risk. At some point you should enable it again to configure it so that it doesn't hinder reaching your application from outside.