Fake or genuine IWC GST chronograph or not?

There is more fishy than the date, e.g. the reference number: IW 5004-01 is a big pilot.
The serial number indicates that it indeed is a fake. Google it and you'll find several links to fake watch communities stating exactly your serial number.

Have you checked if all of the pin release buttons on the bracelet work? I've seen some very sparingly accurate fakes of the GST chronograph but on all of them the buttons only worked on two or three links around the clasp.

Unless they changed the boxes over the years... it also doesnt look right. But from the card alone, stay away.

The reference number on the warranty card is indeed from a Big Pilot. This is very strange. I would go with my fellow WUS members above.