Curl & onPaymentNotification()

In order to empty the cart, you need to have the line below in the onAfterOrderConfirm function:
$this->removeCart = true;

And HikaShop will handle that for you.

If it still empties the cart even though you don't have that code in the onAfterOrderConfirm function, I suppose that it's because you don't have the latest version of HikaShop.
Prior to that, the cart was always emptied when the order was created. We now moved the decision on whether to remove the cart or not to the payment plugins so that you can do just that.

Mmm. When you submit a form, the browser goes to the URL defined in the action attribute of the form tag. So it depends where that goes. However, you should be able to trigger again onAfterOrderConfrim once a checkout has been finished by just redirecting to the URL:
index.php?option=com_hikashop&ctrl=order&task=pay&order_id=XXX
where XXX is the order id.