Mark,
1) No, you can't synchronize the comments between a status update on your Facebook Page or Profile that links to an article on your site and the comments attached to the article itself. The comments made on a status update (even if it includes a URL to your site) would be for the status update itself..perhaps you made a positive or negative comment about the article, etc, but the they aren't necessarily related to the article itself. So, Facebook keeps those comments separate.
We'd love it if there was a way to synchronize them in certain cases, but there is no way that we know of to do so and have never seen that for other sites either.
2) When a user makes a comment and Posts it to Facebook, the comment will post to the user's wall. With that post, there will be a link to the original article as well as the comment itself. From there, the user can follow the links and post a comment back to the user if they want to continue that thread.
3) You can't hide the "Post to Facebook" button on the comment box. It's up to the user if they want their comment to show in the (possibly public) wall/timeline. Some comments, they may just want to appear on a website in a more private fashion or simply because they don't feel the comment is relevant to all of their friends. It's a choice for the user, not the administrator.
4) I'm not sure what you mean here, but if you need more clarification, just let us know.
5) No, we don't recommend auto-posting articles to Facebook. There are other extensions that do it, and we've seen a lot of problems with those extensions including early-posting (before the article is published), double-posting, or incorrect information about the actual article. We always recommend using our Open Graph tags to 'describe' the article how you want it to look when it's shared on Facebook by specifying the title, description, and image tags. Then, using the status update feature of Facebook to create a good intro to the article that's tailored to your Facebook users. If you simply auto-post the first sentence or 2 of an article, it can be very impersonal. Creating a comment that will draw your Facebook users in to actually read the article has proven to be much more effective. It may take a few minutes more effort, but it is a much better way to social engage your users and allows you to sculpt the look and timing of the post.
Hope that answers all of your questions, but if you have others, just let us know!
Thanks,
Alex