If you take a picture on Snapchat and discard it without sending it, putting it on your story, or saving it to memories, can it come up again and does Snapchat store it?
When you take a picture on Snapchat, it is then stored for approximately 10 seconds after which is removed. There has been a claim that the deleted pictures can be recovered, but that requires forensics knowledge. It's not something that can be done by the regular users.
I recommend reading the policy information provided by the Snapchat team themselves: https://www.snap.com/en-US/privacy/privacy-policy/