I am looking for references / hints of proof on the derivation of the Euler characteristics $\chi(\mathcal M_{0,n})=(-1)^{n-1} (n-3)!$ of the moduli space of rational $n$-pointed curves.
I have been trying to show this by using recursion and gluing rules on boundary divisors, but (i) I have to apply a bizarre selection rule for the boundary divisors, (ii) I am wondering if I am not making a conceptual mistake by mixing up $\mathcal M_{0,n}$ and its Deligne-Mumford compactification.
This is just a sketch, and probably needs a little detail-filling in.
For the sake of laziness, I'm going to write $M_{0,n}$ instead of $\mathcal{M}_{0,n}$.
You can think of $M_{0,n}$ as the space of punctured curves. Moreover, you have natural forgetful maps $$ M_{0,n+1} \to M_{0,n} $$ given by forgetting points. What is the fibre of such a map? Well, it is just the (punctured) curve itself! Consequently, its Euler characteristic can be computed to be $\chi(C,x_1, \ldots, x_{n+1}) = (2 - (n+1))$.
So playing around with the fact that Euler characteristics on fibrations are multiplicative...
Anyhow, this isn't rigorous, but it should give you an outline to work with.