is it not possible to allow user decide their own user name

On this site, we're using the setting of "Create New Users: No". This automatically creates a psuedo-Joomla account for the user with the fb_xxxx username.

If you use the setting of Yes (which is what's used on the demo site below), the user is allowed to choose their own username, full name and other account information. As the admin, you can select which fields are imported from their profile, but this is irregardless of their privacy settings as when a user grants permission to use your app through Facebook Connect, they are allowing you to access their profile data.
Demo site: facebook.sourcecoast.com

If you have suggestions on how to change these features, we're all ears. At one point, we had it so you could set the auto-generated username. However, we removed that feature due to confusion and issues. Each username must be unique, so using the Facebook ID of the user guarantees this, and hasn't been an issue in the past.

Certainly, if you'd like us to change your username on this site, we'll gladly do it. Just submit a contact-us request.

Thanks!