Path: /system/ajax StatusText: Not Found

404 sounds like a case where clean urls is not working ok. Please check if that is working correctly.

If not, could also check the patch in #1519916: Field templates, field refresh and cogwheel problems ?

Also, in case it doesn't work wit DS disabled, then you have a complete different problem.

I am also experiencing the second issue ("Invalid form POST data.") - also with 7.x-2.6 as well as 7.x-2.x-dev.

I have clean URLs enabled and they are working fine. I tested with Google Chrome to see if perhaps /system/ajax (or /en/system/ajax in my case, since I am running i18n) caused 404 errors when trying to move a field around, but they did not. I have no way to test this without i18n - it is an absolute must in my case and I just have this one dev server for this project. However, I was running i18n before with Display Suite 7.x-1.9 without any problems for the past two years on a production site. For what it's worth the production server was running Ubuntu while this new dev copy is running CentOs, so it could well be down to Apache config differences. I did also try this:

<IfModule mod_security.c>
  SecFilterEngine Off
  SecFilterScanPOST Off
</IfModule>

(from https://www.drupal.org/node/1944464#comment-7221412)

I did also apply the patch from https://www.drupal.org/node/1519916 manually ... no effect.

Edit: I looked into ajax_get_form() which is where that error message originates... it calls this:

$form = form_get_cache($form_build_id, $form_state);

I double-checked that $form_build_id is populated correctly (it is). $form_state is populated with an array which looks correct to me as well. I do not have any caching enabled on this site right now, not the native caching nor Varnish.

Edit: upon further testing I now see this issue only happens on some content types, perhaps due to certain field types.

This is usually the problem when you don't have mod_rewrite extension for Apache enabled. Run a2enmod rewrite to enable it.

Thanks
Yes the problem with a2enmod rewrite. I enable it and restart apache.