What was the conclusion with regards to the reformation of the body when splatted then? The specific protein identifiers is an interesting idea, but I feel as though at that point, the cells themselves would be destroyed, preventing complete reconstruction of the body. In blood cells, the protein identifiers are used as a means to identify non-self cells that enter the body, such as bacteria, that eventually leads to destruction of the foreign body, leaving nothing intact. This theory, while fairly sound, seems to ignore that possibility from this summary (I'll go back through the thread eventually). From what you wrote, it sounds like an enzyme that would have allosteric inhibition from the identifier proteins of friendly inklings that prevent the destruction of those with the correct identifier. Would the ink itself really need to have the identifier proteins then?
I am quite mad at myself for not knowing about the sea cucumber liquefaction part as I am studying marine biology right now. From what I have read on this, it is neurological control over collagen in the tissues that allows this. It seems as though the cells themselves never really separate, though. Some articles even describe that the near total liquefaction of certain species would likely cause death, and it seems like this more complete liquefaction is what inklings are able to do. I'm not really sure how to amend this outside of complete separation of the cells with some sort of advanced, specific signaling that would follow, likely hormonal or chemical and related to the identifier proteins. This really pushes it though, whereas the sea cucumber bit at least has a natural example.
For the organs, could it be possible for them to simply split while going through a grate? It could work in a similar way as to endo/exocytosis where the walls of the organs form a "vesicle" that surround the part of the grate, allowing them to pass through and remain intact. This may not work for the heart and circulatory system, but other invertebrates may have a solution for that. Most invertebrates (cephalopods being that only mollusks that don't) have open circulatory systems with hemolymph instead of blood. In these systems, the blood leaves the vessels and floods the tissues to provide them with oxygen. If inklings were to have reverted to an open circulatory system, at least in squid form, then the issue of the blood vessels being blocked while passing through grates is resolved, but not the heart. Either the heart could stop beating for the short time it takes or instead of using an open circulatory system, it could revert to simple diffusion again. While inefficient, the inklings only exist in this form for a short amount of time (squid form). Both options are a stretch, but they could also offer options to leave the fourth dimension away from the organs. It might not be a good idea for part of the inkling to begin existing in another dimension while the rest doesn't.
For the ink in the fourth dimension bit, are you saying that the ink itself is a four-dimensional object? From my understanding of the dimensions (which I admit can be lacking), three-dimensional objects/beings (like us) exist in four dimensions. So for the ink to have a fourth dimension to help explain the organs' phenomenon, it would exist in five dimension? (Not trying to refute anything with this one. I just think its really cool.)
Overall, the conclusions described seem well put together. Sorry if some of it sounds like nitpicking, it's not meant to be offensive in any way if it comes off that way. I'll be sure to read through the rest of this thread as well. Yay for walls of text.
(Big) Edit: What if aforementioned enzyme breaks down collagen if the squids liquefy in a similar manner to sea cucumbers? This could explain why enemy ink readily liquefies inklings while it takes a short transformation time to do it by choice. Food for thought. Also, after a bit of reading the previous pages, I support the idea of a modified hydrostatic skeleton, possibly combined with the evolutionary remnants of the pen, which could have become primarily made of collagen. For the rapid evolution aspect, it could be a combination of the theory of punctuated equilibrium and hazardous waste released into the environment by humans. I mean, if rapidly changing environments lead to rapid evolution, I think a globe-encompassing flood fits the bill.