How to display properly exception or echo string after posting in plugin?

As per the comments, using global variables is the suitable solution as you are seeking to store custom messages from the headers which will be displayed later with the markup, and using sessions would just result in extra code and extra memory usage (performance) while you can keep things simple.

If you print any message from your class, it will be printed on the top because yet WordPress hasn’t started output.

Try this:

public function saveParametre()
{
    global $my_plugin_message;
    try {
    #... Do something ...
        if ($var) {      
            $my_plugin_message="Sucess";
        } else {
            $my_plugin_message="Failed";
        }
    } catch (Exception $e) {
        $my_plugin_message="Erreur: ".$e->getMessage()."\n";
    }
}

Now check for the global var $my_plugin_message while outputting your form:

<!-- form elements -->
<?php global $my_plugin_message;
if ( isset( $my_plugin_message ) ) : ?>
  <?php echo esc_attr( $my_plugin_message ); ?>
<?php endif; ?>
<!-- .. -->

That’s one of the many advantages of global variables. Glad it helped.