I think a similar technique should work for surfaces in general, the idea being to use the Mayer-Vietoris sequence to convert a question about the dimension of $H^1$ into a question about the dimension of some $H^0$'s which counts connected components, which can in turn be detected by numbers of square roots.

Yes, the pullback of a form from the quotient is invariant, so you need only check that an invariant form is the pullback of a form. Given an invariant form on $X$, one can define a form locally on $Y$ away from the ramification locus by defining it as the value on one of the covering sheets. It is easy to check that this form extends holomorphically to all of $Y$. For instance, using Riemann-Hurwitz (or really just the local description of holomorphic maps $z\mapsto z^n$) to verify that the extension is holomorphic.

Thanks for the reply, though I think I lack the necessary background to see why condition 2) was satisfied in the example described. I will think about it more, but if you could point me in the right direction it would be greatly appreciated.

Because $\Gamma$ is a subgroup of a discrete group $O^+(L)$ of isometries of lattice, it is discrete. Whether the group $\Gamma$ is geometrically finite may not be that helpful, because I would like to quotient $A$, not $\mathbb{H}^n$, by $\Gamma$. Furthermore, geometric finiteness is not sufficient to conclude the existence of a polyhedral fundamental domain in dimension $4$ and higher.

Two fibers of the projection $\mathbb{P}^1\times\mathbb{P}^1\rightarrow \mathbb{P}^1$. I will make my comment an answer, to elaborate.

Maybe I'm missing something, but the inverse images of two disjoint fibers are two disjoint surfaces in the blow-up. The images of these surfaces in $\mathbb{P}^3$ must then only intersect at the four points, which is impossible, as two surfaces in $\mathbb{P}^3$ always intersect in at least a curve.

Interesting example, but as Francesco's answer shows, this cannot occur over $\mathbb{C}$, which I should have mentioned was the setting of this question.