Server caching (possibly Varnish?) interfering with AJAX Webforms submit

I actually don't know what the problem is, exactly, but here are the known factors:

1) I have a dev and staging environment that are more or less identical, running latest stable version of Drupal 7. The Drupal codebase is the same.

2) Here's the code I'm using in MYMODULE:

MYMODULE_form_alter(&$form, &$form_state, $form_id) {
  if ($form_id == 'webform_client_form_TARGETNID') {
    $form['actions']['submit']['#ajax'] = array(
      'callback' => 'MYMODULE_ajax_callback',
      'wrapper' => str_replace('_','-',$form['#form_id']),
      'effect' => 'fade',
    );
    $form['#submit'][] = 'webform_extra_ajax_submit';
  }
}

function MYMODULE_ajax_submit(&$form, &$form_state) {
	$form_state['rebuild'] = TRUE;
}

function MYMODULE_ajax_callback(&$form, &$form_state) {
  if(form_get_errors()){
    drupal_get_messages();
      return $form;
  }else{
    return $form['#node']->webform['confirmation'];
  }
}

3) The above code works wonderfully in dev. It worked in staging, as well. Then I began installing various server-cache solutions. I'm using Ubuntu 10.04, and pretty much all the parts of the LAMP stack that I can think of have simply been installed using apt-get install, so nothing fancy, and versions are whatever you'd get from doing that. In fact, staging was started as an image of dev two months ago, with appropriate changes to hostname, /etc/hosts, and /etc/apache2/sites-available/MYSITE.

4) I installed apc using "pecl install apc":

echo "extension=apc.so
apc.shm_segments=1
apc.shm_size=128M
apc.ttl=7200
apc.user_ttl=7200
apc.enable_cli=1
apc.stat=1
apc.stat_ctime=1" > /etc/php5/conf.d/apc.ini
apache2ctl restart

5) I installed memcached following the steps from this page:
http://www.zayinkrige.com/installing-memcached-on-ubuntu-10-04/

6) I installed Varnish:

• /etc/varnish/default.vcl: http://pastebin.com/5bFhmVZ0
• updated /etc/apache2/ports.conf:
  

  #NameVirtualHost *:80
  #Listen 80
  NameVirtualHost *:MYPORT
  Listen *:MYPORT

• updated /etc/apache2/sites-available/MYSITE:
  

  #<VirtualHost *:80>
  <VirtualHost *:MYPORT>

7) I updated settings.php and downloaded/installed the APC, memcache and Varnish Drupal modules:

// Add Varnish as the page cache handler.
$conf['cache_backends'][] = 'sites/all/modules/contrib/varnish/varnish.cache.inc';
$conf['cache_class_cache_page'] = 'VarnishCache';
// Drupal 7 does not cache pages when we invoke hooks during bootstrap.
// This needs to be disabled.
$conf['page_cache_invoke_hooks'] = FALSE;

// this could be very good, but need to do more research, first.
// $conf['page_cache_without_database'] = TRUE;

$conf['cache_backends'][] = 'sites/all/modules/contrib/apc/drupal_apc_cache.inc';
$conf['cache_class_cache'] = 'DrupalAPCCache';
$conf['cache_class_cache_bootstrap'] = 'DrupalAPCCache';
//$conf['apc_show_debug'] = TRUE;  // Remove the slashes to use debug mode.

$conf['cache_backends'][] = 'sites/all/modules/contrib/memcache/memcache.inc';
$conf['cache_default_class'] = 'MemCacheDrupal';
$conf['cache_class_cache_form'] = 'DrupalDatabaseCache';

# Tell Drupal it's behind a proxy
$conf['reverse_proxy'] = TRUE;
# Tell Drupal what addresses the proxy server(s) use
$conf['reverse_proxy_addresses'] = array('127.0.0.1','MYSERVERIP');

8) Expected behavior (works on dev, and worked on staging before APC/memcached/Varnish):

9) Observed behavior (even after I've done my best to reverse as many of the changes for APC/memcached/Varnish that I've described, above):

Conclusion:
I want caching, but I also want to figure out why this won't work. Am I not turning all these off correctly? My biggest suspect here is Varnish, but maybe it's something else? I'm not even sure how to troubleshoot this at this point. Any suggestions would be very much appreciated.

Thank you!