add following code in your theme’s (child theme’s if any) functions.php

add_action( 'fep_display_before_directory', function(){
	?><script type="text/javascript">
	jQuery(document).ready(function($){
		var delay = (function(){
		  var timer = 0;
		  return function(callback, ms){
		    clearTimeout (timer);
		    timer = setTimeout(callback, ms);
		  };
		})();

		$('.fep-directory-search-form-field').keyup(function() {
			delay(function(){
		      $('#fep-directory-search-form').submit();
		  }, 1000 );
		});
	});
	</script><?php
});

for messages, add following code in your theme’s (child theme’s if any) functions.php

add_action( 'fep_display_before_messagebox', function(){
	?><script type="text/javascript">
	jQuery(document).ready(function($){
		var delay = (function(){
		  var timer = 0;
		  return function(callback, ms){
		    clearTimeout (timer);
		    timer = setTimeout(callback, ms);
		  };
		})();

		$('.fep-messagebox-search-form-field').keyup(function() {
			delay(function(){
		      $('#fep-messagebox-search-form').submit();
		  }, 1000 );
		});
	});
	</script><?php
});

Not possible.

It depends on your website speed, this code has nothing to do with that. It only submit your form without click.