Where can I see that a customer has requested a Gift Message, Wrap or Receipt?
Whenever a customer is selecting a certain option from the Gift Options pop up modal, the cart attributes are being updated "behind the scenes" with the option or options that the customer has chosen.
While adding a Gift Wrap is visually adding a new "Gift Wrap" product to the cart, the other two options - Gift Message and Gift Receipt can be seen by the store owner (you) in the "Additional Details" section of each order:
The location of the chosen Gift Options of an order
Now you'll be able to process with the modifications that the customer has requested.