Thanks for your interest and for making me think, thinking is my favourite pastime... although I don't do it very well, sorry...
Yet, I will try to answer your queries.
A) Counting and math are, seems to me, inseparable. The development of math was to a large extent due to our capacity to use abstraction and use symbols representing numbers. Thus, math, as you say, is a "language of logic", and as well the art of counting/quantifying (forgot who said this or where I read it, somehow I liked that expression because, as I describe in the text, seems to me all started with counting).
B) I am not sure there is a "right" coordinate system, as you mention there are several to choose from. My point was that if we eliminate those coordinates and focus on the important thing, the relative positions of the elements, this may be beneficial in terms of simplifying. Simplification is what I am after (in my personal and professional lives!).
C) You are right in that if we study or characterise interactions/relations among elements, some "coordinates" are needed. So even if we get rid of cartesian or polar or any other coordinate, there is still the relative position, namely the separation between the possibly interacting elements, that will determine whether or not those elements will interact (exchange energy). Hence one cannot get rid of space, that is, some sort of "coordinates", because after all, space is all there is. Some metric, then is needed. But I would prefer the simplest metric that does not use absolute reference frames, rather the relative positions, although, naturally, the relative positions have to be estimated with something, so here we are back to some sort of coordinates?... I don't know. In any event, I liked Merleau-Ponty's definition of space as the means whereby the position of things become possible; it is a more dynamic view than the classical one and helps conceptuliase together the "position/arrangement" of things with the progression of their interactions/changes, that is, with time. To my mind, this perspective helps me understand that space and time are, like someone said, modes by which we think and that these two are completely inseparable, two sides of the same coin. so to speak. Although bear in mind I am not a physicist, so this sort of high-level, abstract thinking does not come easily to my mind, I am just an experimentalist scientist interested in the nature of that thing we term reality.
D) I would hope that, as you say, the distinct perspectives are translatable into each other. And indeed, the simplest perspective would be the, let's say, most convenient! Not sure what kind of example you mean, that you were trying to find in the essay. I could try to guess what you were after, so let me give you a simple example. This is not really a math versus other, simpler view, it is an instance of a very complex thing becoming simple at higher levels of description. There is a wide variety of biochemical alterations that give rise to the many epileptic syndromes, but going "beyond" molecules and focusing on the collective activity of neurons during epileptiform events, and analysing this collective activity using math (of course!), particularly dynamical system theory, we can come up with a relatively simple description that can be described in few words as to why epileptic seizures occur and even how we can stop them. Once I read that Feynman said something like a good theory should be one that is understandable without the need to solve the equations. So going back to the beginning of this paragraph, the complex mathematical view and the perhaps simpler view (although I suspect it would be of a mathematical nature too, but hopefully simpler) should be compatible and translatable into each other, but maybe we don’t need to “solve” all the equations to comprehend whatever natural phenomenon.
E) I tried to stay away from these metaphysical aspects, and besides, as you also mention, it is a matter of semantics. These are terms we created and that help us communicate, but we should not take them too seriously, for these are our subjective notions. What I mentioned about the properties and things/structures was needed to make myself understood that relations between properties is what we almost always perceive, what most scientists work on. I would also admit that properties are all we sense, and never the essence of things. As far as I know, nobody has ever seen a protein, or an alpha particle, all we have "seen" are their properties, light diffraction or dispersion, bubbles in a cloud chamber we attribute to the particle, etc.
F) The crucial point is that about "somehow everybody gets money and be it from social welfare...". In my career I have witnessed how, to my opinion, excellent scientists had to quit academia because the money they were getting was not enough.... although it was enough for them but not for their institutions!! Such is the nature of academia these days, more like a corporation, and if they judge you are not obtaining enough moneys, they will invite you to leave, with tears in their eyes of course! Once you leave academia, well, it may not be that easy to perform research, other priorities appear: jobs, etc. Your Einstein example is a valid one, but I am afraid, he was the exception that proves the rule. Regarding the big money flowing into science, you may be right in that the result is a clog in the system. One reason is that the big moneys are normally awarded to big groups producing tons of megadata. Today we have such a flood of data that, in fact, it is clogging us up. My colleagues and I have devoted thought to these matters (published a few blogs, texts...), thinking about alternatives to remedy the immense avalanche of data and make sense of all these observations by a different money distribution policy, but to tell you all this would take a long, long time.