BuddyPress.org

Did you install any social plugin? I guess not..else you would have known that the login with buttons are created automatically by the plugin…try the following social login plugin

https://wordpress.org/plugins/wordpress-social-login/

Regarding your main idea..you would need to do some custom stuff for that…as far as approval of users go use the following plugin…

https://wordpress.org/plugins/bp-registration-options/screenshots/

googling really helps though