Casimir effect, Lamb shift, Rabi vacuum oscillations, electron
anomalous g-factor... The quantum vacuum is brimming with stuff.
Zitterbewegung,

Structure. Brimming with structure, which is ordinarily invisible due to a
relativistic effect. But it is definately there.



Then when I think of time, I think of a man-made logical device used
for measuring the "distance" between the occurrence of two events of
interest. (Time seems to be one of those things you can't define easily
without using the word "time" itself, though I am aware of the
definition of a second in terms of transitions of state in the
cesium-133 isotope).


Time is the 4th dimension.



Furthermore, how can space curve? That sounds like taking "nothingness"
and giving it shape which doesn't seem to compute in my mind.

Empty space is not nothingness. It is merely an invisible substance. It is
just dimension, but it's definately not a "nothingness".




OK,
suppose that it could curve and we treat it as a hypothetical fourth
state of matter coming after the gaseous state. Wouldn't it need to be
contained in something and hence take on the curved shape of that
something (much like a liquid in a container)? Or looking at it another
way, how would curved space maintain its structure?


I believe that waves in spacetime may exhibit some type of superfluidity.
This is why atomic particles are so stable. Consider a photon travelling
across the whole universe. The photon is just a wave in spacetime, and yet
this wave has the ability to traverse the diameter of the universe. The
photon is a wave which is propagated through the medium known as spacetime,
and it is a "frictionless" process.

Spacetime waveforms exhibit superfluidity.