You were mapping the email provider field to email CB field. This is a core field that's already mapped. Trying to map it from configuration can cause a conflict. I've removed that mapping and tested your site with my personal Facebook and it worked perfectly fine. The avatars are synchronizing, but you have approval enabled for avatars (you can disable this within CB Connect). You may also want to review the permissions you're requesting carefully to ensure you actually need all of those.

If you're still having issues with your account try deauthorizing the application within your Facebook account from your privacy settings. This will allow you to reauthorize the application to see if that could be the case.

You have written:
"Trying to map it from configuration can cause a conflict. I've removed that mapping."
Where have you removed that mapping?

As explained above you were mapping the email field. Mapping is done in CB Connect > Parameters > Facebook > Registration. Core fields are automatically taken care of by CB Connect and should not be mapped (this will be fixed in a later release, which will prevent anyone from even doing that).

Only when user manually insert the email adress it works.
But why is this field not mapped correctly?

So avatar, etc.. all work fine for you? This could mean your Facebook account simply is not providing email address. Could be a permissions problem in your account it self. Email address is not guaranteed and is why there's the email request page if it's missing.

I have no idea whats wrong. Please help.

Me either. I've used my own Facebook account on your site and it worked perfectly fine. Your Facebook account is not providing the requested information and I've no idea why. Do you have another Facebook account that you could try? Navigate to Facebook > Settings > Apps and find your application. Next click edit and see if the email permission is given and what email address was given to the application.

As you requested i checked with another Facebook account (newly created for this this). With this account it works.

Then the issue is specifically with your Facebook account. I don't know what Facebook isn't providing an email address from your account though. You can try to contact Facebook to see if there's a permission somewhere for that, but beyond that there's nothing I can do. CB Connect is using the API in accordance with their documentation and works with my account as well as your newly created one.