After my patent application is approved by the European Patent Office, is it valid only for that designated country or for all EU contracting states?

Nowadays, when submitting a European patent application, all member states of the European Patent Office are designated. The application itself is valid for each country, unless you withdraw the designation of a certain country.

However, whether the patent is valid in a certain country after authorization depends on whether you have paid the relevant fees and registered in that country.

Even if a European patent is authorized, it is still in name only. It needs to be registered in a country before it can be valid in a country.