We can see things moving faster than light

The early statement "the galaxy, *right now,* is 32 billion light years away" is an entirely meaningless statement. Simultaneity doesn't work at a distance. This is relativity 101.

You can easily see things moving faster than _c_ by looking at a star or even the moon and whipping your head around. Ah, but that's rotating, so it's cheating, I hear you cry. No, rotating frames of reference are perfectly fine in the General Theory of Relativity. That's why it's *General.* A corollary  is that the _artificial gravity_ such as in a rotating space station like in _2001_ is just as real as the gravity on the Earth. In fact, locally, it's identical from the Equivalence Principle. At larger scales, it's just that the metric is different. 

Relating all this stuff to Euclidean geometry and our common-sense notions of motion, position, and time, are fine if they help you understand something, but they usually make it more difficult. Brains just aren't configured to be able to understand large distances, speeds, times, and so forth and so on. They work pretty well _locally,_ and Einsteinian relativity is a purely local theory (plus the assumption that it's still true everywhere and everywhen, which I think is adequately supported by Noether's Theorem.

Almost all confusion about physics I conjecture is due to the fact that we use brain structures that are there to help us understand our immediate environments. Confusion over the two-slit experiment, for example, can be directly traced to brain structures rather unfortunately called _topographic maps_ which are there for purposes like figuring out if a fleeing animal is going to go to the left or the right side of a tree.