Update: The vanishing order error has been resolved, too.
In our case, it was caused by the "Order Status" setting in the "PayPal Payments Standard" config.
The option "Store Default Order Status" caused the order to be deleted after checkout. There's no reason why it does this. There are no "delete"-queries anywhere near the checkout_process/checkout_success-scripts.
We set the "Order Status" to "Delivered". This seemed to fix the issue.