Status Message Contrib Modules Don't Work

Did you place the messages pane in your Panels Everywhere site layout configuration?

Aha, so you want to use a contrib module that does some stuff with the messages based on JS? I would guess that due to us overriding the theme function for the messages there is some incompatibility. I am going to blame those modules for that because of how they attach their JS functionality then. If it breaks by changing some minor stuff in a template or theme function there is something wrong with the JS.

I'd suggest you take a look at the JS provided by the module and see how you can override it, or, alternatively, use status-messages.theme.inc in your theme and invoke theme_status_messages() from there (to override Omega).

I don't know about the other modules, but Better Messages overrides the theme function (to add markup) which in turn gets overridden by Omega. I couldn't find a simple way to restore the registry's value to before Omega changed it. So for the sake of using just Better Messages, create a file "status-messages.theme.inc" in your sub-theme's "theme" folder and put the following inside (changing MY_THEME to your theme name):

function MY_THEME_status_messages($variables) {
  if (module_exists('better_messages')) {
    return theme_better_messages($variables);
  }

  return omega_status_messages($variables);
}

I'm using Disable Messages and have the same problem.
Messages that should be disabled are still visible if my Omega theme (but not in Bartik or another theme) is set as default.

I too fixed it by doing this:

<?php
/**
 * Implements hook_theme_registry_alter().
 */
function SUBTHEME_theme_registry_alter(&$theme_registry) {
  if (module_exists('disable_messages')) {
    $theme_registry['status_messages']['function'] = '_theme_disable_messages_status_messages';
  }
}
?>

#7 fixed it for me, thanks malberts!

I'd suggest #7 over #8 because it does not break the inheritance chain / chain of overrides. Under normal circumstances, #8 is just as good as #7 but it can get ugly if you have other theme registry altering functionality in place. E.g. I am pretty sure that if you wanted to use sth. like devel_themer, the registry altering approach would break it.