Interesting question. Maybe. Considering that the stores INSIDE Disneyland park give the Signature AP 20% merchandise discount (whereas the DtD one does not), makes me think that the stores inside the park are Disney owned.
You can also buy Pandora at one of the stores on Main Street and that is DEFINITELY a Disney owned store and you can use a gift card there. They have a smaller selection than the one in New Orleans Square, but if they have what you are looking for, you can use your giftcard.
Yes. We have done so multiple times.
There are two types of employees in the Pandora shop in New Orleans Square. The Pandora employees answer questions about the items you want and help with fitting, etc. When you finish making your selection from them, they turn you over to a Disney cast member who actually rings up the purchase. That includes any typical Annual Pass discounts and charging mechanisms like Disney Gift Cards.