Does it make sense to view all solutions to Einstein's field equations as $g_{\mu \nu}=\eta_{\mu \nu}+h_{\mu \nu}$, for a given $h_{\mu \nu}$? I have a question about General Relativity that so far I have never been able to solve.
Let's take Einstein's field equations:
$$ R_{\mu \nu}-\frac{1}{2}g_{\mu \nu}R=\frac{8 \pi G_{N}}{c^3}T_{\mu \nu}. \tag{1}$$
By imposing proper initial conditions and an energy-momentum tensor, one is able to find solutions for the metric tensor that include the Schwarzschild's metric, the Freedman-Robertson-Walker one and so on.
Now, let's define the following
$$g_{\mu \nu} \equiv \eta_{\mu \nu}+h_{\mu \nu}\tag{2}$$
and let's insert this expression into the previous Einstein's field equations.
My question is the following

"Given a proper $T_{\mu \nu}$, are all solutions in $g_{\mu \nu}$ and $h_{\mu \nu}$ equivalent?"

Is it possible that there exist certain solutions that cannot be expressed as a Minkowski background $\eta_{\mu \nu}$ plus fluctuations $h_{\mu \nu}$?
 A: 
Is it possible that there exist certain solutions that cannot be expressed as a Minkowski background $\eta_{\mu \nu}$ plus fluctuations $h_{\mu \nu}$?

No. Take any arbitrary solution of the Einstein field equations, pick a coordinate chart, and define
$$h_{\mu \nu} := g_{\mu \nu}-\eta_{\mu \nu}.$$
Then the metric on that chart is given by
$$g_{\mu \nu} = \eta_{\mu \nu}+h_{\mu \nu}.$$
However, there is no guarantee that this is at all useful, that $h_{\mu\nu}$ is at all simple or small (over anything more than a local approximation on a chart chosen to be so small that the deviations don't have time to come in), that this will be possible globally instead of over only a limited chart, or that you're doing anything other than pure symbol-pushing.

Is it possible that there exist certain solutions that cannot be expressed in a useful way as a Minkowski background $\eta_{\mu \nu}$ plus fluctuations $h_{\mu \nu}$?

Yes, absolutely. This is the generic case - most metrics are not reducible to a perturbed Minkowski background. If you want a concrete example, start with the Schwarzschild metric.
A: 
Is it possible that there exist certain solutions that cannot be expressed as a Minkowski background $\eta_{\mu \nu}$ plus fluctuations $h_{\mu \nu}$?

Yes. There are links between curvature and topology, and some topologies are not consistent with a Minkowski metric. For example, closed FLRW spacetimes have the spatial topology of a 3-sphere, which is not compatible with a Minkowski metric.
Even for spacetimes that topologically could admit a Minkowski metric, there would be the question of what coordinate system to impose on the spacetime in order to identify some coordinates with the Minkowski coordinates. In general this coordinatization is not unique.
A: I think you didn't understand perturbation theory in the context of GR. You split the metric tensor into an exact solution ($\eta$) of Einstein's equations, and a perturbation ($h$) as:
$$g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu}$$
If you substitute this into your equation $(1)$, you get an infinite series in $h$, on both sides of the equality sign. One then typically chooses to ignore perturbations of order higher than one, given they are negligible compared to the background ($\eta$ here). This is known as linearized GR. One can choose to stop at any order of perturbation they like, but then equations get messy quickly.
If $g_{\mu \nu}$ was an exact solution, you wouldn't split it into (background + perturbation) in the first place.
Also, you cannot fix $h_{\mu \nu}$ from the start. $h_{\mu \nu}$ is the dynamical variable in the perturbation theory you construct, and it is exactly whose equations of motion you need to solve for.
Sometimes, perturbing around flat space is not ideal, like above. For example, when dealing with perturbations around a Schwarzschild solution, you want to perturb around the exact Schwarzschild solution in GR. For instance, see this classic paper by Regge and Wheeler - Stability of a Schwarzschild Singularity. This gave birth to black hole perturbation theory, the fruits of which we are witnessing today from gravitational wave observations.
A: You may also be interested in the existence of the Kerr-Schild exact solutions to GR. These allow to have space-times with potentially large curvature, and treat them in a way that's similar to what you have in mind when you say "useful". You start decomposing the metric as $g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}$ with $h_{\mu\nu} = \phi k_\mu k_\nu$, such that $\phi$ is a scalar and $k_\mu$ is a null vector for both $g$ and $\eta$. The metric is easy to inverse, you just pick up a sign in front of the "perturbation", and the Einstein equations linearize in terms of $\phi$. Many black holes spacetimes can be put in this form. For Schwarzschild, you'd have $h_{\mu\nu} = \frac{2GM}{r} k_\mu k_\nu$ with $k^\mu =(1,x^i/r)$, $r=x^i x_i$, $i=1\dots3$.
