I have connected to my remote server using the FQDN machine name: server1.mydomain.com, one day later I cannot any more connect using server1.mydomain.com but I am still able to connect using the dynamic public IP address. Is there something to configure to make RDP follow dynamic dns?
This has nothing to do with your server needing to follow dynamic DNS or having the DNS role installed.
When the IP address changes, it takes time for your computer to realize the address has changed because it caches the results of DNS queries. You can clear this cache by running
ipconfig /flushdns from an elevated command prompt.
Also, DNS servers between you and your dynamic DNS provider can cache their responses. Your dynamic DNS provider can set the length of time that downstream DNS servers are supposed to cache DNS entries, but not everyone obeys those values. Therefore, you could be at the mercy of their cache configuration.