Few questions -
1) Is it possible to create private forums only for specific users/members?
2) If so, how? And where to set who has access?
3) We're using WishList member as a membership plugin and give access to LifterLMS courses. They're integrated to access to certain level=> access to a certain course.
Is it possible to link access to a specific (private) forum with course enrollment?
You should create a specific usergroup for that specific user. Only usergroups can be restricted to access some forum. Please read this blog post: https://gvectors.com/how-to-set-up-private-forum/