re: If I login via the widget, it leads me to the index-page but not to the directory
Generally the page seen after logging in will be the one the user is already on. If this is not happening it is likely because another plugin or theme has added a login action which overrides the login destination. I recommend reading these two FAQs:
re: If I’m going to be registered as a new user, it leads only to a “castrated” WP-admin page. The settings that I can change there have no influence to my connections profile. Even if I click on “profile” via the widget after logging in, it leads only to the wp-admin profile. I expect to come to my own Connections profile
The user’s profile is the user’s profile in WP. The login widget does not change this. The user’s profile is not a directory entry. They are designed to be completely separate.
re: what I absolutely need is, that every user can set up his/her own Connections profile (add entry) after the registration.
You need the Link extension. Your users will still need to registered. After they register they can add and maintain their own entry in the directory.
re: By the way, an administrator has to approve it first when someone new does a registration. How does this work?
When you setup the Link extension you can chose if the user’s submission is to be moderated or not.
Hope this answers your questions thoroughly!