The ability to post answers anonymously

Anonymous mode might be useful for this. For details about how it works, see: Enable and configure anonymous mode. If this is for a site that’s on our hosting, get in touch with our support team if you would like us to enable anonymous mode for you.

With anonymous mode enabled, to only allow anonymous users to reply to questions that have been created by an admin, you’ll need to configure a category so that its security settings allow admins to Create / Reply / See and allow the ‘everyone’ group to Reply / See. For details about how to set this up, have a look at How to use category security settings to create private categories.

Enabling anonymous mode might not achieve everything that you are wanting to do. For example, anonymous users would be able to post in other categories on your forum. It would be possible to prevent this by using custom groups and category security settings, but it could take some work on your part to set it up.