Your problem might be related to wrong permissions, eZ publish installation wizard needs write permission (e.g 770) to var/ settings/ and design/ directories. It looks that proper configuration files wasn't created during installation. Try set correct permissions to such files, check also issues with Finetune button (if exists).
If you are new to eZ publish world I suggest to take a while and read this tutorial after installation: http://ez.no/products/ez_publish_cms/documentation/building_an_ez_publish_site
It will help you understand basics and you will get knowledge how to work with eZ publish templates.
I set the permissions to 777 (all until main folder).
After setting thes permissions I already get the warning: Warning
open_basedir
open_basedir is in use and can give problems running eZ publish due to bugs in some PHP versions. It's recommended that it is turned off if you experience problems running eZ publish.
In addition to this it is recommended to run the php.ini in a safe mode. Is this really necessary?
Now I always see the Finetune-screen nothing else. The same finetune screen is shown when I choose "Ignore this test"
Can you please give me advice one´s more or do have any more ideas to solve the problem?
eZ publish will not work with PHP in safe_mode. Ask your administrator for help. He can disable it for eZ publish main directory. This should solve your problems.