Hi,
looking at your website from the frontend i can't determine the problem with the Facebook login because there are no error messages, but only a silent missing login. Maybe having an admin access and investigating i could find something more.
About Linkedin the problem is due to the old php version running,probably still PHP 5.3
Linkedin login requires at least PHP 5.4 so you must update it or ask to the hosting provider.
My direct email address for further communications is
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Cheers!