There are two errors in the package.
The installation has only worked after I added an "Extended" to the Contact.controller.php (-> ExtendedContact.controller.php) and on the page ExtendedContact.integrate.php was also an "Extended" missing.
I replaced there
$actionArray['contact'] = array('Contact.controller.php', 'Contact_Controller', 'action_contact');
with
$actionArray['contact'] = array('ExtendedContact.controller.php', 'Contact_Controller', 'action_contact');
After this the installation was successful and apparently all works fine now.
Oh, just one more thing.
Besides the input fields for "Message" and "E-Mail Address" another one for "Name" would be great.
How can I add such a field there?
EDIT :
Now the contact form just works when I'm logged in. Guests always will be forwarded to the index page, after klicking the contact button in the menu...