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Is this like... spoiling the game? Or.. how does the game work? Can you pull up a bunch more pictures of them easily? Do you own these kitties?

No I do not own them. A detailed guide can be found here: https://coinsutra.com/cryptokitties-cryptocollectibles/

You can have a replica of all that exist to show off. a big Family photo album for sure. LOL
Cheers