Is it possible to exchange session data between ajax-calls and normal request? (in WordPress)

Solution:

Global keyword should be inside the function you want to use it in, like so:

public function get_current_formvalues() {
        global $wp_session;
        return $wp_session['formvalues'];
}

And also add it in the set method

 public function set_formvalues() {
         global $wp_session;           
         //Default values for form

The way it is currently written, $wp_session is not availble when called. You can read more on http://php.net/manual/en/language.variables.scope.php