Hello!

Is it possible to enable new Users manually?

So the Administrator can check who is allowed to log into the Forum who not?

And after that:
Is it possible to "open" the Forum only to the logged-in Users? So nobody else can see the Posts?

1. See this option

//$registerInactiveUsers=TRUE;

in the manual.

Another way is to make all forums for registered users only, to disable users' registration and they to ask you via e-mail or contact form to register them manually.

2. Read THIS. Hope it will help you.

Read this and search our forums before posting answered questions.