>> So the automatically installation of a local database through the script will configure it ready for use for the gluu server?
Yes.
>> If i want to use an separate remote database server what configuration are necessary to the database-configuration for example for MySQL to use it for the gluu server?
You just need to install a vanilla mysql server in that remote server and open port for Gluu Server, that's it.
>> Does the gluu server installation also needs an own reverse proxy (like nginx or haproxy)?
No.