Can Space-Time Itself Have Energy Qualities Like Momentum?

I'm going to build off of chris's original comment. I haven't had a response from him during the 24 hours since I asked the question, and there doesn't appear to have been any activity from him recently (Note: don't confuse him with another user who goes by Chris), so I might as well expand upon what he said.

Gravitational waves do appear to be what you're looking for. They are emitted by systems with varying quadrupole moments (see https://en.wikipedia.org/wiki/Gravitational_wave and https://en.wikipedia.org/wiki/Quadrupole for more information); commonly cited examples are binary neutron stars. In fact, one such binary, the Hulse-Taylor binary, was the first discovered system to emit gravitational waves.

Gravitational waves carry energy away from the system, at a rate of
$$frac{dE}{dt}=-frac{32}{5}frac{G^4}{c^5}frac{(m_1m_2)^2(m_1+m_2)}{r^5}$$
where $E$ is eneryg, $t$ is time, $m_1$ and $m_2$ are the masses of the objects in the system, $r$ is the distance between them, and $G$ and $c$ are the constants, the universal gravitational constant and the speed of light. I invite you to do the calculations for a given system, if you please. I can assure you that it's one of the easier calculations in general relativity! This release of energy causes the orbits of the two neutron stars to gradually decay, and it is thought that eventually the two will merge.

The answer boils down to this: Yes, gravitational waves can carry [angular] momentum, just like many other types of waves. They also have frequency, amplitude, wavelength, and speed, just like the "normal" waves we are familiar with.

I hope this helps.