I'm not a lawyer; if this extends beyond idle curiosity, go see one.
... have I created something materially 'new' to which I now can claim my own (limited?) copyright?
You've created what's called a derivative work, which means you've used a substantial part of an existing work as the basis for creating something different. The changes -- and only the changes -- are included in the copyright you hold when the dust has settled. For example, if you draw a mustache on a photo of the President of the United States, you hold copyright on the mustache but copyright on the original photo is still held by the photographer.
The hitch is that you must have the right to use the original to create the derivative. If you don't, use of the existing material is unlawful and you receive no protection for the derivative work.
Is this analogy from an answer to another question relevant in this case? ... RAW files are just that — unbaked data. You wouldn't go to a bakery and demand that you get the flour, sugar, and eggs with your cake...
Not really. Even in the other question, it falls short because none of the ingredients of a cake are creative works. More importantly, the ingredients can't be copied: making another of the same cake requires different eggs and flour even if they're turned into a cake the same way. The finished cake may be a creative work, but that's another discussion.
A raw image isn't particularly attractive to view in its unprocessed state, but it does contain data that represents a creative work. It's no different than if you'd published a straight-out-of-the-camera JPEG because you thought it was perfect. If I license a processed image to you, the license only covers that image. The raw is a different work not covered by the license.
Do I have a (if any) right to post my processed version of the photographer's original RAW image without having to request express permission?
For that raw image, yes. As was pointed out elsewhere, anything in /r/EditMyRaw is posted under CC BY-NC-SA 4.0. That license give you the right to make and distribute changes as long as its terms are obeyed.