Michael,
1) Tables remaining after installation - That's very intentional. It preserves configuration settings and Facebook->Joomla mappings so that if you install again, that information isn't lost. We don't have an automated way to remove those tables right now. It's not optimal to leave tables hanging out, we agree. I can see making a 'cleaner' tool.. one day.
2) The checkmark for the component is intentionally missing. We thought about it right before release, but the check just tells you if the listed extension is published. Since you only see the extension check in the component, it's unnecessary to say the component is there.. we agree though that it should be there to avoid this question in the future and will likely be added in soon
We'll look for the cause of those messages and see if they're real or a duplicate message again. As for redirection, if you have "Automatically log Facebook Users In" enabled, and are having issues with the login redirection, that will be fixed in the 3.1.2 release coming early next week
If not, ask away!