Neat, almost nothing about me except four "likes", and two of those is for a Facebook app I've made.

Neat. Not sure what the following means (I saw it for "Outbox" and a few others):

{ "error": { "type": "Exception", "message": "You can only access the \"outbox\" connection for the current user: " } }

Does that mean I have to be logged in to get at that data?

You can only access graph.facebook.com/me/outbox if you're logged in and it's only accessible for yourself.

To access those you need to get authenticated and it's only allowed for the logged in user.

The graph api pretty much displays what you will see at your vanity url www.facebook.com/<username> without being logged in or when a user who is not your friend comes to your page.