While it would slightly clutter the board, requiring people to get the Best Ending in this would be absurd - so other difficulties would absolutely have to be an option. I'd probably add them as people actually do runs in each difficulty.
As for if one somehow accidentally picks up a non-required that doesn't actually impact game play - such as the Ogre Medal - it shouldn't invalidate the run.
As for the others, it would absolutely be a niche category. I can't say I've much interest in it myself, but it is an interesting challenge.
If you or someone else does a run of it, I'd be happy to add it as a category.
Other 100% runner chiming in here, also asking that 100% be added. It's a very fun category that shows most of what Cave Story has to offer. Has some pretty stressful moments, specifically Ironhead, making it quite entertaining to watch as well.