Why Can We Observe Space Curvature / Warping At All? I don't understand why we are able to see and measure curvature / warping of space at all.
Space as I understand it determines distances between objects, so if space were "compressed" or warped, shouldn't distances be compressed or warped the same way (like crumpling up a sheet of paper) ?
Then, however, our units of measure and frames of reference should be compressed likewise so that there should not be any visible changes to our cognition.
This would also rule out warp drives unless they form some wormhole in hyper-space (analogous to two points touching each other on the crumpled sheet of paper)...
What am I missing ?
 A: 
I don't understand why we are able to see and measure curvature /
  warping of space at all.

The Earth's surface is curved and this can be observed via the vast number of pictures of the Earth from space that now exist.
However, the surface curvature can also be "seen" via measurements on the surface itself.
For example, if one were start at the North Pole and travel in a "straight line" (a great circle) to the equator, then move east along the equator for a quarter of the circumference, and then move North (always along a great circle), one would eventually reach the starting point at the North Pole.
But look, one would have formed a "triangle" with the interior angles adding up to 270 degrees!  This is one way that intrinsic curvature is measured.
Simply put, intrinsic curvature is mathematically characterized by the Riemann Curvature Tensor and observed via geodesic deviation.
A: Curvature affects how objects in the universe move and interact with one another, and these effects can be measured.
Take, for example, the phenomenon of gravitational lensing.  Because spacetime curvature can deflect the path of light, we can potentially observe light coming from objects that are directly behind other objects.  Here's a nice picture.
As another example, planets orbit around the sun and each other because of spacetime curvature, and we can certainly measure these effects.  Specifically, we can measure the period of orbit of a certain planet and compare that to what general relativity tells us that the period will be given that the sun curves spacetime in a particular way.
When we make measurements of curvature, we're not literally going to some point far out in space, putting some clocks and rulers there, and "measuring" how much spacetime is warping by counting seconds and ticks on the ruler.  We're making predictions about how the curvature should give rise to certain phenomena, and then we're checking those predictions.
A: General Relativity deals with curvature of Spacetime, not just curvature of Space. You can't ignore time because clocks are affected throughout the universe. Spacetime events are what we measure and are independent of observers.
Now, let's come to point: You're asking why we're able to measure effects of Spacetime curvature with classical way when reference frames are also affected with Spacetime curvature.
Simple Answer: Any specific Spacetime curvature doesn't affect everything uniformly throughout the universe. Mars isn't affected equally as our reference frame on Earth is affected by Sun (ignoring curvature due to Earth and countless other things for simplicity).
