Is it possible to access Facebook APIs for logging in a user, or to allow a user to become a fan of a fan page through a third party site, without actually creating an app on facebook?
Why don't you create a fab page directly for users to become fans for that page? And as far as i know, you can not access facebook API without API key.
You can't access the API without creating an application and getting an API key. There are a small number of API queries that will work without the user having added an application or even interacted with it but they generally on retrieve very basic data.
There's certainly no programmable way of making a user a fan of a page without using a fan box widget or having them do it directly through the site. This is because the post_form_id value which Facebook uses for these kinds of interactions is only present on pages served from facebook.com and is never accessible to the developer. To allow otherwise would open up the system to all kinds of exploitation.
I've no idea what problem you're trying to solve anyway. Creating an application API key is no hardship (and it doesn't have to be an active application for session-less queries - just create an app and grab the keys) and the fan box widget can be styled with CSS, as long as you follow the basic rules that Facebook has set out.