Belyi's theorem states that if a Riemann surface could be defined as an algebraic curve over an algebraic number field, then this Riemann surface could be described by a Dessin d'enfant. I have two questions:

Is there a way to describe the moduli space of genus $g$ bordered/punctured hyperbolic Riemann surfaces with $n$ borders/punctures using dessins d'enfants?

Let's say that we want to compute an integral of the form $\int_{M_{g;n}}[D\tau]\,\,f[\tau_1,\cdots,\tau_{6g-6+2n}]$ over genus $g$ bordered/punctured hyperbolic Riemann surfaces with $n$ borders/punctures in which $[D\tau]$ is a suitable measure. Mirzakhani has computed the volume of the moduli space. But what if we have a more general integral? what is the best parametrization? Fenchel-Nielsen coordinates? I suppose that it should be case-dependent.