Where does gravitational waves' energy go?

To a first approximation gravitational waves are never dissipated. They just spread out into the universe gradually getting fainter.

Gravitational waves are exceedingly difficult to dissipate for the same reason that they are exceedingly hard to generate in the first place. They couple very weakly to matter. The gravitational wave detected by LIGO stretched than compressed the Earth by a factor of about $10^{-21}$. The Earth is squidgy, meaning that if you stretch it then let it relax you don't get as much energy out as you put in - the rest goes to heating up the Earth. So in principle some of the energy in the gravitational wave was dissipated as it deformed the Earth. However in practice the fraction of its energy that the wave lost is utterly insignificant. Possibly the bits of the wave that hit Jupiter and the Sun lost a bit more energy, but remember that most of the wave passed through the Solar System without hitting any matter at all.

However gravitational waves do get fainter with time for two reasons. Firstly the gravitational wave from a black hole merger propagates roughly in a plane so its intensity falls off as a factor of somewhere between $\frac{1}{r}$ and $\frac{1}{r^2}$, where $r$ is distance away from the source. Secondly the energy in the wave is diluted as the universe expands. Actually the expansion not only dilutes the energy but it also red shifts it, so if $a$ is the scale factor by which the universe has expanded the energy of the wave falls as $\frac{1}{a^4}$.

You ask:

If impossible, does the universe end up with nothing but GW?

but this isn't going to happen simply because it's so hard to produce gravitational waves. The matter currently lying around in the universe is mostly going to remain lying around in the universe for the foreseeable future.

Actually gravitational waves aren't unique in not being dissipated. Light interacts very strongly with matter, but most light emitted by objects in the universe isn't going to hit any matter simply because the universe is mostly empty space. For example most of the photons emitted in the cosmic microwave background (CMB) haven't hit anything in the 13.8 billion years since, which is of course why we can still see the CMB.

With time reversal symmetry you should be able to spin up two black holes if you play back the gravitational waves onto them. The Lagrangian of GR does not explicitly depend on time, so it should be invariant under time reversal, I think.

That way there should be a possibility to absorb gravitational wave energy back into the system.

In any case the energy (or power) gets distributed on an enormous surface. The energy (or power) must decay with $1/r^2$ if it is a spherical wave. As you have astronomical distances there, the amount of energy reaching earth is very little.

And I think you would have to hit some resonance frequency in order to actually absorb the energy. The frequency of most violent gravitational waves is probably not the right one to set the earth in motion. In most cases the energy will just move through us.