Check user logged in with JS

Just use wp_localize_script() and is_user_logged_in().

wp_localize_script( 
    'my-ajax-request', 
    'MyAjax', 
    array( 
        'ajaxurl' => admin_url( 'admin-ajax.php' ),
        'logged_in' => is_user_logged_in()
    ) 
);