My non-DRM books do download data from Amazon. As long as the books contain the same ASIN it downloaded the page numbers in my experience.
Regarding the Fire and apps, It should be the same book unless you have converted it anyway, since the Kindle must decrypt the book in the same way before opening it. It should keep the same internal super-secret Store markup after the deDRM up until you convert(/probably editing as well?) and therefore Amazon should think it IS their book. It would be equivalent to downloading from MYK.
Keep in mind I have the e-ink Kindle Touch, and have no idea how any other device might or should work, except by guessing.
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