Interesting representations/cohomology of surface groups? For purposes of my own, I'm interested in constructing connected spaces, without recourse to geometric realisation or the like, that have non-trivial homotopy groups in dimension 1 and 2 and are not products of Eilenberg-MacLane spaces. This rules out obvious constructions using crossed modules (at least, obvious to me). 
One idea is this: take a representation $\rho:\Gamma_g \to SO(3)$ of $\Gamma_g = \pi_1(\Sigma_g)$, the fundamental group of a compact, connected, orientable surface or genus $g$. Then form the associated sphere bundle $X=\widetilde{\Sigma_g} \times_\rho S^2 \to \Sigma_g$. 
Then, unless my calculation is wrong, $X$ is a 2-type space with $\pi_2(X) = \mathbb{Z}$, $\pi_1(X) = \Gamma_g$ and $k$-invariant $a\in H^3(\Gamma_g,\mathbb{Z})$.
So my question is, 

are there any non-trivial representations $\rho$ or cohomology classes $a$?

and secondarily,

would a non-trivial representation $\rho$ give rise to a non-trivial $k$-invariant in the above situation?

 A: As Torsten Ekedahl mentioned, the resulting object will not be a 2-type.  For this, you would instead want to construct $\mathbb{CP}^\infty$-bundles over your surface.  These are classified by homomorphisms Γg → ℤ/2, determining the action of $\pi_1$ on $\pi_2$, together with an associated cohomology element (which is zero for the dimension reasons stated in the comments above).  There are therefore (ℤ/2)2g different 2-types with these homotopy groups.
If you really did want to construct S2-bundles, it turns out that (again because surfaces are low-dimensional) these are in bijective correspondence with 2-types under the map that sends such a bundle to its second Postnikov stage.  As a result, these are again determined solely by how the fundamental group acts on $\pi_2$.  Only the trivial bundle arises from an SO(3)-action because SO(3) acts trivially on the first homotopy group of S2.
There are many distinct representations of the group Γg (e.g. g=1!) and they will obviously not necessarily give rise to distinct bundles.  One reason for this is that the associated bundle only depends on the path-component of the representation $\rho$ in the space of all SO(3)-actions (or homeomorphism-actions, or self-homotopy-equivalence-actions).
A: For the first part of your question, there are faithful representations $\rho: \Gamma_g \to SO(3)$. One way to see this is by taking an arithmetic realization of the surface $\Sigma_g$, giving a faithful rep. $\rho':\Gamma_g \to PSL(2,R)$. Then a non-trivial Galois conjugate of this representation will give a faithful rep. $\rho:\Gamma_g \to SO(3)\subset PSL(2,C)$. 
