at.algebraic topology - Branched coverings over orbifolds with reflector lines

A detailed write up of what you are looking for is also available in Chapter 13 of Thurston's notes. Specifically, formula 13.3.4 on page 331, has the desired formula
$$chi(O) = chi(X_O) -frac{1}{2} sum_{i=1} ^N (1-frac{1}{n_i}) -sum_{j=1}^M(1-frac{1}{m_j}),$$
where $O$ is the orbifold, $X_O$ is it's underlying space, there are N points fixed locally by dihedral groups of orders $2n_1,...,2n_N$ and $M$ points fixed locally by rotations of orders $m_1,...,m_M$.

However, I believe the question addressed (at least in part) by Hurwitz in the proof of Hurwitz's theorem, since the bound the order of an automorphism group H acting on surface of genus g is 168(g − 1) if H includes orientation reversing symmetries and 84(g-1) in the orientable case. The key observation is that the Gauss-Bonnet extends to orbifolds as well.

Computing the Euler characteristic for the quotient of $mathbb{H}^2$ by the (full) (2,3,7) triangle group is a good way to see this in action.