Ideal Ford domain for finite index subgroup

Question

Let $G$ be a lattice Fuchsian group with parabolic elements, seen as a discrete subgroup of matrices $ g= \begin{pmatrix} \alpha & \overline{\beta} \\ \beta & \overline{\alpha} \end{pmatrix} $ with $|\alpha|^2-|\beta|^2=1$, acting on the Poincaré disc $U$. I am interested by the property (P) below.

Property (P) There exists a finite index (free) subgroup $G_0$ of $G$ whose Ford domain $D$ is an ideal polygon, that is it has all its vertices on the boundary of the disc.

The Ford domain $D$ of $G$ is the complement in $U$ of all isometric circles of $G$. The isometric circle of $g$ (not fixing the origin) is the circle with centre $-\overline{\alpha}/\beta$ and radius $1/|\beta|$.

Question 1 Does (P) hold for any Fuchsian lattice $G$ with parabolic elements? Are there counterexamples?

Arguing as in Theorem 5.4 of https://arxiv.org/pdf/0911.4957.pdf one can see the following fact.

Fact 2 Let $Q$ be an ideal polygon with finitely many sides, such that a side $L$ is a diameter of $U$ (that is it contains the origin). Let $H$ be the group generated by the reflections in the sides of $Q$. Let $G$ be the index $2$ subgroup of orientation preserving elements of $H$. Then $G$ has Property (P) with $G_0=G$.

Question 3 Can you find a larger class of groups satisfying (P)? (for all groups as in Fact 2 the quotient $U/G$ is a punctured sphere).

Finally I resume from https://arxiv.org/pdf/0911.4957.pdf some elements of the proof of Fact 2.

1. All angles at vertices of $Q$ are equal to $0$, thus $H$ is discrete. Hence $G$ is discrete too (section 9.8 in Beardon, "Geometry of Discrete Groups")

2. Let $\sigma_L$ be the reflection in the diameter $L$. Let $e_1,\dots,e_n$ be the sides of $Q$ other than $L$. Let $\sigma_j$ be the reflection in any such $e_j$. Set $D:=Q\cup\sigma_L(Q)$ and $s_j:=\sigma_L(e_j)$ for $j=1,\dots,n$. Then $g_j:=\sigma_L\circ\sigma_j$ are generators of $G$ pairing the sides $e_j$ and $s_j$ of $D$. Thus $D$ is a fundamental domain for $G$.

3. Any $g_j:=\sigma_L\circ\sigma_j$ is the composition of the reflection in the circle $e_j$ and the reflection in the line $L$. Thus $e_j$ is the isometric circle of $g_j$. Similarly $s_j$ is the isometric circle of $g_j^{-1}$. Hence $D$ contains the Ford domain of $G$. But both the Ford domain and $D$ are fundamental domains. Thus $D$ is the Ford domain of $G$. Such Ford domain is an ideal polygon.

Thanks for any insight on any possible extension of Property (P)!