In his paper "Weil-Petersson Volumes" Penner gives the following formula for the integral of a top-dimensional cohomology class $\omega$ on the moduli space $\mathcal M_g^s$ of $s$-punctured riemann surfaces of genus $g$:

$$\int_{\mathcal M_g^s}\omega=\sum_G\frac{1}{|Aut(G)|}\int_{D(G)}\phi^*\omega$$

where the sum is taken over all isomorphism classes of ribbon graphs $G$, $\phi:\widetilde{\mathcal T_g^s}\to\mathcal M_g^s$ is the natural map from the decorated Teichmüller space to the moduli space, and $D(G)$ is more or less the pre-image under $\phi$ of the orbicell in $\mathcal M_g^s$ corresponding to $G$.

My question is to what extent can this formula be generalized. Specifically,

1) Is a similar formula for evaluating forms of arbitrary degree?

2) Is this formula the ``shadow'' of a general formula for evaluating cohomology classes on (smooth) orbifolds?

I myself cannot understand Penner's paper well enough to see where the top-dimensional assumption on $\omega$ enters the picture. References to the literature would be much appreciated as well.