Managing comments and anonymous posts

1) The comments are all stored on Facebook. There is no copy of them locally in your database.
2) No, the user must be logged into one of the services. This is a Facebook requirement, and there's no way around it.
3) You can manage them locally. If you are an administrator of the Facebook Application, and logged into Facebook, there should be a "Moderate" link under each comment where you can hide it or do other things that normal user's can't do. If you just added the comment box, it may take Facebook up to 48 hours to recognize your Application ID for the comments box and start adding the moderate button. If it doesn't appear, let us know, and we can help further.
4) JFBConnect doesn't have it's own comment system. There are many great comment systems out there for Joomla, and others integrated into different blogging or article components. We'd recommend you use one of those if you'd like to have a fully on-site conversation.

Hope that all helps,
Alex