Is there any possible way to study the finite-dimensional, irreducible representations of $Diff(R^4)$ systematically? My interests stems from the fact, that the symmetry group of general relativity is $Diff(R^4)$.
To make what I have in mind a bit more precise: The symmetry group of special relativity is the Poincare group. Its finite-dimensional, irreducible representations can be studied systematically by looking at the Lorentz group first, and noting that its Lie algebra can be seen, speaking colloquially, as being built of two copies of the Lie algebra of $SU(2)$. The representations of $SU(2)$ can be constructed in a straight-forward way, and therefore the representations of the Poincare group, because the Poincare group is the Lorentz group plus translations. By systematically I mean, that the representations can be labeled with the corresponding values of the Casimir operators and its possible to start with the lowest possible value and then move the ladder upwards.
Any reading suggestion or idea would be much appreciated.
First, the symmetry group of gravitational fields is not the diffeomorphism group of Minkowski spacetime. In fact, for many solutions to Einstein field equations, the corresponding diffeomorphism group of spacetime are infinite dimensional Lie group with many components. On the other hand, take the anti-de-sitter spacetime as an example, its symmetry group in $D>3$ is simply the conformal group in $D-1$, which is finite dimensional.
You may be interested in the palatini approach to general relativity in which the Einstein gravity is re-formulated as a gauge theory. In this formalism, the metric of spacetime is not treated as fundamental objects in our theory. Instead, knowing the metric of spacetime, we study the frame bundle over spacetime with gauge group $SO(1,D-1)$. In this way, general relativity as a theory studied on tangent bundle of spacetime is reformulated on the frame bundle over spacetime. In this formalism, metric is not fundamental, but we choose a set of orthogonal frame, called vielbeins, and the connection on this principal bundle, called spin connection. Then, general relativity is a sort of $SO(1,D-1)$-gauge theory.
Anyway, the symmetry group of a manifold should be found via solving the killing field equations and this has nothing to do with the whole diffeomorphism group itself.
For the representation of Poincare group, its representation depends on your interest. If you are doing classical fields, then it is given by the spinorial representation as you described. If you are doing the quantum mechanics, then it is given by unitary representation so that the transition amplitude is preserved. Since Poincare group is non-compact, its unitary representation must be infinite dimensional. You may also consider some other interesting representation in which the spin number is continuous, although we have never observed such particles in nature.