What Step Would You Need To Prove A Coin Is Real

If I understand you correctly, you've got two separate questions here: "How would you authenticate a coin that mintage figures says does not exist?", and "How would you authenticate a piece of scissel?".

The first question: the short answer is, "with great difficulty". You would need several things: provable provenance going right back to the date that the coin says would be a great help. You would need some indication or theory as to why, even with zero mintage, at least one coin was minted anyway and some evidence or testimony to back this up. You would also need to allow experts in the field of which your coin is a a part to examine it, as closely as they wish for as long as they wish, in order for them to come to a meaningful conclusion.

In the case of a "1937 penny", you'd need to come up with some explanation as to how a die that was only made and used for uniface patterns somehow got attached to a normal coin press with a normal obverse die and at least one coin was struck using that pair of dies. You'd also need to allow the experts to match your coin with the known 1937 patterns; since only one 1937 reverse penny die was made, it must match - or you must come up with some explanation for the creation and subsequent disappearance from all records of a second 1937 penny die.

The second question: scissel (the scrap metal left over after a coin blank has been punched out of it) was not supposed to leave the mint; it was supposed to be recycled. Any that left the mint probably left illegally, making it technically stolen government property. It would be difficult, therefore, to get good and proper "provenance". That being said, if the size of the holes was correct for penny blanks (not actual pennies) and metallurgical analysis proved the composition matched that of the penny, then you could with some confidence say that it is indeed likely to be a piece of penny scissel. But I don't think complete, absolute certainty is a possibility.