Hi,
We have gone through the issue and as per my understanding the orders are not correctly updated after the payment is correctly done, also the status is not correctly updated as well so it does not appear in the dashboard as well.
We have worked on many prestashop website and are very well know to the structure of prestashop, so we know the complications which we are going to face in this issue.
We are ready to start the work and look forward to your response.
Regards