I think when it says "claim photos" that's for when you got pictures taken but didn't scan a band, but got a photo card instead. You would click "claim photos" and enter the code from the card to make your account populate with those pictures.
I personally like to download a set of pictures with no edits before starting to add borders and effects, because the system is a bit slow and cumbersome and I want to make sure I get a clean copy of everything.i think if you don't choose "save a copy" when you're editing, it just changes your original and you're only left with the edited version. After I download one set, I edit and add borders or stickers then download those edited photos.