If you enable “login required”, then absolutely nothing will be available to the public. One thing that might work for you is:
- Make all your categories accessible only to people in a certain group, e.g. “team”
- Add all of your staff to the “team” group
- Disable “login required”
- Make a single “FAQ” category which is readable publically
You’ll have to be very careful when creating a post/category that it ends up with the correct permissions, but it might be workable.