As documentation I have only found 1 old YouTube video. Is there any more documentation about how to get the new version of CB Connect up and running?
There isn't any new documentation yet. I have yet to find the time to redo the tutorials. They are still somewhat valid though and can be found below.
www.joomlapolis.com/support/tutorials/99-plugins
HybridAuth also provides some tutorials that you can find on the below page.
hybridauth.sourceforge.net/userguide.html#index
Now I want to have the Facebook tabs and I am looking for the special Connect field-types (like facebook social plugin) as mentioned in the video but I cannot find them.
Those don't exist anymore. The social plugins provided by social sites are basically just HTML and can be added using Custom HTML field.