Curvature of spacetime as a real thing? 
*

*I get the curvature tensor in General Relativity, it is “just” math. Does space-time REALLY curves as a tangible thing, or is Einstein proposing a mathematical abstraction? 

*More naively, please allow, is space-time a real physical “something” like a “new ether”? If yes, does anyone have any idea what is it made off? 
 A: Theoretical physics

"Theoretical physics is a branch of physics which employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experimental tools to probe these phenomena."

Obviously since Einstein was in fact a theoretical physicist, that is what he did. 
General relativity makes many different predictions, such as gravitational time dilation and gravitational redshift, and these have all been confirmed in observation, that is good "evidence" that general relativity is correct. As is with all scientific theories. There is a myriad of evidence and tests that support GR, such as gravitational lensing in which you can read about here.
There are different groups of individuals who believe spacetime is a substance that exists independently of the mass-energy within it, the other group thinks spacetime is defined through spatiotemporal relations between matter in the universe. Simply put, whatever the "fabric" of spacetime truly is, is a mystery, but it exists nonetheless as something and curves/bends in the presence of mass-energy and momentum.

A: This is going to be more philosophy than physics, but I feel it might shed some light on the discussion.
Philosophy gave us our original ideas of time and space.  The conceptual framework of Newtonian physics - itself based upon Descartes' unification of Euclidean geometry and algebra - makes use of these two foundational metaphysical ideas.
We may think of those concepts are "intuitive" but don't forget that our everyday languages embed such metaphysical notions and subliminally condition us to swallow these [now proven wrong] concepts of time and space.  Hopi is an example of language that doesn't have the same "metaphysical" assumptions of time vs. space that we do.  ( http://en.wikipedia.org/wiki/Hopi_time_controversy ) 
Now... on to Einstein et al... the reason we consider him a genius is that he discovered that those eminently reasonable - on the surface - metaphysical assumptions of "time" and "space" actually FAIL to hold when you think deeply and subtly enough.  Einstein saw that the emperor wasn't wearing any clothes.
However, Einstein did not come up with a more intuitive metaphysical concept to replace those broken ideas of "time" and "space". Instead, he glued time and space together using hairy - for the lay person - mathematical constructs.
Most of today's physicists, while they have no problem dealing with "spacetime" in their equations, are STILL generally held hostage to the bogus /metaphysical/ ideas of linear time and three-dimensional space (and most "educated" laypeople even more so).  I believe they fail to underestimate just how much their use of language affects their perception.
To think more clearly about it, we have to realize that:
1) "time" that does not "flow uniformly" SHOULD NOT BE CALLED TIME at all!
2) "space" that "warps" SHOULD NOT BE CALLED SPACE because it isn't, not in the original sense!
It's like the paradox of "what happens when an irresistible force meets an immovable object?"  The simple answer being that this is a FAILURE OF LANGUAGE.  By definition, there is no such thing as an irresistible force if there is such a thing as an immovable object, and v.v.
Paradoxes, it seems, are an indication of the use of broken language/concepts.  So these "paradoxes" we have in physics such as retrocausality, etc... are the result of insistence on clinging on to broken metaphysical concepts.  And we cling to those because /we don't have anything better yet/. 
Before Copernicus, Ptolemaic astronomers used the elaborate mathematical model of epicycles to describe the movements of heavenly bodies.  It worked to a certain extent, just like our physics theories today, but epicycles were unwieldy and cumbersome.
The math became vastly more simplified once they threw out the one bogus axiom: that the Earth was at the center.  Most likely, as Planck says, these older astronomers had to die out first.
My radical suggestion is that even today's leading-edge physicists - despite what the math keeps showing them - still refuse to throw away certain bogus axioms, and is what gives rise to all the complexity of our physics theories today ( http://amzn.to/1IBVk6M ).  These bogus axiomatic beliefs are that:
1) There is a "past" and a "future".  E.g. A Feynman diagram achieves its symbolic elegance from doing away with those two ideas.
2) That there is such a thing as "space".
It may sound like we are venturing into crackpot territory here by suggesting the elimination of those two ideas, but realize that Copernicus was definitely considered a crackpot in his time:  
400 B.C. - Zeno's paradox calls into question the validity of the idea of space/distance
20th Century - Einstein's tested/proven theories of relativity mean that the idea of space is bogus (there is only "spacetime" - possibly a huge kludge)
Early(?) 21st Century - NASA successfully develops a warp drive ( http://www.nasa.gov/centers/glenn/technology/warp/warpstat_prt.htm )
... and yet we still buy into the notions of "distance", "location", "past", "present" ?  One day, belief in those ideas will be regarded as having the same naivete as the belief that the earth is at the center of the universe or Aristotle's ideas of "natural place" and "natural motion".
