Lisa,
I just realized that you're looking at the admin area of your site for the Activity Log. Since those actions are published, we don't allow you to delete them. The record of the action must remain in the database so that the user can delete the action from their Timeline. We also don't let you delete it so that you aren't altering (deleting content) from the user's Timeline, that's something they should do.
Once an action has been deleted by the user on the front-end, or the action was never published due to an error, you can delete it from your logs in the admin area. Hopefully, that makes sense. Better description text back there explaining why you can't delete them would probably be nice. I can see that an "Archive" button might be useful to at least hide some actions, if you'd want to, as well.
Home Page Open Graph
Your home page still does have a query string, it's just not necesarily shown in the URL. The same would go for an SEF URL.. there is an underlying query string you can still use. To do so, go to the Menu Manager and click into your home page menu item. There is a parameter for Link, which you can't edit. That will have the query string variables in it. You can click, drag, highlight and copy/paste that to notepad or something and use those values for the custom tag. It would look something like:
index.php?option=com_content&view=featured
From there, you should be able to create the Custom Object as normal.
Facebook Amex Big Break
Congratulations! We saw that contest going on and almost entered, but were pretty sure our work with Facebook would disqualify us. I'm excited to see that we're playing a part in your site, and that award, in the long run. Very well done, and we hope you the best of success. If you have to present to Facebook, feel free to drop our name... a lot
It's exciting to see what you, and a few other venturesome folk, are doing with the Custom Open Graph Actions. We will have easier implementation of those features over the next few months, and really appreciate your feedback so that we can tell where the difficult points are. That's how we improve our extension is by knowing what needs work. We know it makes Open Graph easier, because it'd be very difficult without JFBConnect, but also realize there are some rough edges right now.
Best wishes,
Alex