1.) After logging in every user (super user too) gets an alert, that "Error You are not authorised to view this resource.".
I checked module manager and every module set to "public" so that can't be the issue. Could you please check whats wrong?

It looks like you have configured EasySocial to stay on the same page after logging in and since your home page is actually a login layout with "Guest" permissions, hence there is an error message showing up. I have now configured it to redirect to the groups page as per your second question below and it works fine now.

2.) How can I set that after login users get redirected to "Groups" (Csoportok in hungarian) page?

I have published the menu and configured EasySocial to redirect users to the group's page upon logging in, http://take.ms/xRmev

3.) Can I set that after registration every user will be a member of a specific group automatically?
As an admin can I set later to add all users to a group?

You may configure this when you are editing the profile type, http://take.ms/GBu0H. You can define groups for users to automatically join when they register on the site depending on their profile type.