What happens to gravitational waves after arbitrarily long propagation? Given that some systems may radiate energy in the form of gravity waves, and that gravitational waves weaken proportionally to the distance travelled, what would happen to the waves that never hit anything and just propagated infinitely?
Do they just become infinitesimally small? If so, would this imply that energy can exist in a non-quantifiable form?
 A: The amplitudes do become arbitrarily small, and there's nothing at all wrong with this. In fact the exact same thing happens with electromagnetic waves. Sure we have a quantum theory with photons that places limits on how small a packet of energy can be detected, but light can travel across the universe just fine and become as dim as it wants. The intensity of the wave, integrated over the entire wavefront's area, yields the total energy, which is constant.
If you have a quantized theory, all this means is that individual detections will come in discrete packets. If the wave intensity, integrated over your detector's area, is particularly low, all it means is that the number of quanta detected per unit time will be low. The only thing that is quantized (read: discretized) is the energy of the packets, not the energy of the wave itself, the latter being the average energy received over time, whether in packets or continuously.

As a footnote, it is a common misconception that everything in quantum mechanics is quantized. Really the name of the field was poorly chosen. In reality, quantum mechanics is all about linear operators (often non-commuting ones) evolving according to your choice of the wave/diffusion equation. Some of those operators happen to have discrete spectra ("spectrum" means the set of eigenvalues). Discretization is a natural consequence of certain boundary conditions for certain types of problems, but it is by no means the central idea in QM.
A: yes, the waves fall off in intensity as they get farther from the source.  This does not violate conservation of energy, because you'll just be spreading the same amount of energy out over an ever larger volume, but the (energy density)*volume will be constant, minus energy transferred from the waves to matter.
