Is the statement "Claims demand evidence" always true?

[Chas]

I have to inquire, if neurons weren't, in evolutionary terms, designed to track truth

Neurons do no such thing. Nor do brains. Minds do that sort of thing.

-- not for the sake of truth but simply because the more accurate the representation of the world, the easier can harms be avoided and the more can energy-saving advantages be procured -- then what explains our success as a species at overcoming nature and the ignorance she fosters upon us all, especially when this ignorance can be so dangerous?  

What ignorance is that? And I think the word you want is 'foists' not 'fosters'.

Did humans invent the concepts or merely the terms by which to communicate them?  

Of course humans invented the concepts. Those concepts are part of our modeling reality; the underlying reality that exists irrespective of the existence of minds.

Did they invent the "categories" -- of space and time and relation and action -- and the internal consistency that allows us to map our signs/symbols and their theoretical relations onto a world?  

See above.

And through it we have discovered ourselves to be this privileged species, living on a giant ball that orbits around a much larger ball of gas, all of which is in fact less than a spec of dust in the grand scheme of things!  It's all too odd to rule anything out, but it seems less odd to me that the world is as fundamentally abstract as it is physical, perhaps counterparts to the ancient notions of "form and "matter," rather than that the world is only a figment of my mind.  Can one really believe that the only difference in beauty between the Sistine Chapel and some ordinary six year old's finger painting is an arbitrary or irrational judgment formed by one's brain that the former is far more beautiful?

We can objectively determine that the paintings in the Sistine Chapel are statistically less likely that a child's finger painting.
We can objectively determine that those paintings communicate more effectively than a child's finger painting.
We can value them because they are rare and unlikely. We can measure that their emotional effects are greater than those of a child's finger painting.
None of that is arbitrary or irrational.