How can I get the order ID from Commerce (using Rules)?
As you have mentioned, you want to act on completion of the order, so there is a hook of drupal_commerce, which get called on completion..

hook_commerce_checkout_complete($order)


I have used this hook for some action at the order completion in my custom module. If you call this hook in your custom module then $order variable will have all the information related to order. Check this in your custom module --

function MYMODULE_commerce_checkout_complete($order){
  dpm($order);
}