How can I access my database tables? For example when I register a user, I want to change his access rights to admin, but now, without using phpmyadmin I got confused?
Wouldn't you have an admin panel or something where admin can change that using a form? You shouldn't be using phpmyadmin or similar for stuff like that.
Is there a way I can export content to the database?
Sure, look into mysqldump. Basically backup your current database and import it to the forge database.
I created an A record at digital ocean with mydomain.com and pointed to my new server 127.1.2.5, but this does not seem to be the right answer.
I doubt that is the right IP for your server. 127.x.x.x isn't publicly accessible. It's a loopback address, which would only be able to be accessed from the host itself. That refers to the local machine itself and not routeable via the internet.