Let $G$ be a finitely generated Fuchsian group, and let $\mathcal{F}$ denote the Dirichlet fundamental domain of $G$ with respect to $0$ in the Poincaré disc model. Assume throughout that $\mathcal{F}$ is non-compact.
I am interested in properties of $G$, and how these properties are connected with Poincaré's theorem. Here are my questions:
Is $G$ the free product of elementary hyperbolic, parabolic and elliptic subgroups? It is true for the modular group PSL(2,Z). Moreover, it holds under the stronger assumptions in 2).
Assume that $G$ has no elliptic elements. Then $G$ is free (because $G$ is fundamental group of a non-compact surface) and thus, $\mathcal{F}$ has vertices only at the boundary of hyperbolic space. Can we derive this property of the vertices from Poincaré's theorem?
Assume that $G$ is of the second kind (that is, the limit set of $G$ is not equal to the boundary of hyperbolic space). Then in particular $\mathcal{F}$ is non-compact. Is it true that the sides of $\mathcal{F}$ are pairwise disjoint, except (possibly) the sides paired by elliptic elements?
UPDATE: Sam Nead answered 3) in the negative. Is the answer in 3) also negative if we allow an arbitrary reference point for the Dirichlet fundamental domain? Or does there always exist a suitable reference point for a Dirichlet fundamental domain such that the sides are pairwise disjoint, except (possibly) the sides paired by elliptic elements?