A modern view in physics is that when we define a term such as "stationary," we should attempt to do so operationally meaning in terms of some sort of a measurement one can perform. Before we start thinking about the existence of a stationary frame in the real world, we need to define such a frame operationally. Once we have a definition that is sufficiently operationally precise, we can go out into the world and make measurements to see if there exists anything out there that satisfies our definition.
I would recommend that you try to define "stationary frame" in some operational way such that the resulting definition aligns with your intuition for what stationary should mean, and I'll bet you won't be able to do it.
Fortunately, there is another term in physics that is pretty close to what you might want, the notion of an "inertial frame." An inertial frame is one in which if you were holding an accelerometer still in that frame, then the accelerometer would indicate zero. So basically, such a frame is one which is not accelerating. What's interesting about inertial frames, however, is that if you are in an inertial frame, then any other observer moving relative to you with constant velocity will also be in an inertial frame. In other words, if that other observer were holding an accelerometer still relative to him/herself, then he/she too would measure zero acceleration. In such a situation, you might be inclined to claim that you are the one standing still, and the other observer is moving. However, it is a basic fact of physics that the laws of physics are the same in all inertial frames, so from an operational viewpoint, there is no way to distinguish between inertial frames.
It is in this sense that saying that one inertial observer is stationary while another is moving is not really appropriate. It is also because of this that you will often hear statements like "there is no absolute rest" or "motion is relative" or "it only makes sense to talk about relative motion" etc.
The bottom line is that all frames can be categorized as either inertial or non-inertial. All inertial frames are physically equivalent and are the closest you can get to being "stationary" in some sense, while non-inertial frames are pretty far from what most people would call stationary since acceleration is involved.
I hope this helps! Let me know if you want more! (I know I didn't address the questions about distortions etc., but I felt the answer was getting too long).
Cheers!
Addition. Just for the sake of more completeness; it's important to note that infinitely large frames of reference that are inertial don't actually exist. There are, however, "local inertial frames," namely it is possible to devise a physical situation in which a small region around you satisfies the definition of an inertial frame outlined above, but as you move further away from this region, the frame becomes a worse an worse approximation to being inertial. I'd encourage you to explore this further since this point is especially important in GR for example. Here and here would be good places to begin to learn more.
Yes. For example, on October 8th 1970 Earth was in the Sun's radiofrequency shadow with respect to quasar 3C 279. In other words, quasar 3C 279 was occluded by the sun.
Observation from just before and after the occulation permitted measurement of the bending of radiowaves as a test of general relativity.
The sun would also block other frequencies of electromagnetic radiation including visible light.
Best Answer
Here is how you can "run faster than" (or at least, get away from) your shadow: you jump at sunset (I just realized 15 minutes after posting that this is the point that @jkej's answer made as possibility #2)
Your shadow will detach from your feet, and it will "run away" from you. In the frame of reference of the shadow, you are running away from it.
Unfortunately, it won't last... the elephant, in the end, is still right. Unless, of course, you jump just as the sun sets: your shadow would disappear before you land again. Timing may be tricky, but with a good pogo stick you might just do it.
update for @PlasmaHH:
Source of the elephant picture: http://wallpho.com/173361-cartoon-elephant-id-91474.htm