If I set up a subdomain on a domain I manage from a different registrar, how do I set it up in the DDNS control panel? Do I include the full subdomain identifier in the domain name field? Would I redirect my subdomain to the subdomain on your server? Do I need to be a member? Can it even be done?

So to maybe simplify it, how do I setup mysite.MyOwnDomain.com to route through the DYNU DDNS service?

I assume you have some server at home that responds to mysite.MyOwnDomain.com.

What you need to do is the following:
Register your home connection with dynu, and give it a name, for example blabla.mywire.org.
Then, in the DNS for MyOwnDomain.com, you need to make a CNAME record that points mysite.MyOwnDomain.com towards blabla.mywire.org.

This will make any request for mysite.MyOwnDomain.com land on your home IP address.
All you need at home is something that responds to requests for mysite.MyOwnDomain.com.

Regards,
Bert