Hi Jeremy, thanks so much for the response.
The plot thickens! :)
We are having some db issues, and you're right, that could be causing the other issues.
I've also learned we use a hotlink-prevention plugin called Imaguard, which could be causing issues, as well. I'm theorizing I need to whitelist something (simply using facebook.com isn't helping) to get aroudn this.
I've started a separate topic on that point, here:
http://wordpress.org/support/topic/images-not-displaying-on-shared-facebook-posts?replies=1