If Facebook will allow you to access a user’s group membership status, the first project should be possible. You’ll need to assign users to the Discourse group through the Discourse API. The API calls could be made from your WordPress site. Group memberships could also be assigned with a Discourse plugin, but that plugin would not be able to be installed on our hosting.
The second project could be accomplished by customizing the WP Discourse publish_format_html template so that it handles your Facebook feed page differently than it handles other posts that are published to Discourse from your site.