Depends on the product. For some cards at point of purchase a bar code that is on the card itself is scanned. For some it is on the packaging. You might need the packaging in the latter if the card wasn't properly activated.

I throw everything in a bankers box and mark the dates on them. I have plenty of space in the basement to store these boxes so it's no big deal. When I destroy them I use an industrial shredding company as mentioned above. I also throw my other mail with personal info and proprietary work stuff in there.