I've redacted from your initial question the link and login information. I wouldn't post such information in a public forum, especially if you're doing development on the site and need to ensure no one else is coming in and breaking anything.

On your particular site, I gave it a quick look, and you've attempted to use the cart form View for the shopping cart contents checkout pane. This cannot work, because it's attempting to embed a form inside a form. That's why submitting the form does nothing - Drupal just doesn't know what to do! : P

It cannot read the form values passed in to validate the form or determine how to process the form assuming it is valid. You should reset those cart checkout panes to the static cart contents summary.