I have no experience with the Facebook API, so sorry for the newbie question:
I want to write a script to retrieve my contacts and their details from Facebook, just authenticating as myself.
Everything I read suggests the need for an api_key (and I suspect therefore a registered application or something).
Is it possible for me to access my account and its details using the API with just my authenticated user?
No.
Getting your friends (and any of their information) via the Graph API requires an access token. See this page for more details: https://developers.facebook.com/docs/reference/api/
Getting an access token via one of the supported Facebook OAuth 2.0 flows is going to require creating an application because you need the Client ID and Client Secret that’s associated with it. See this page for more details: https://developers.facebook.com/docs/authentication/
The good news is that creating a Facebook application is completely free and there are no strings attached to it, as far as I can tell. Create one here: https://developers.facebook.com/apps and then go have fun building something!