Let $\Gamma$ be a Fuchsian group. Choosing any point $p\in \Bbb H^2$ not fixed by any non-identity element of $\Gamma$, we can construct a hyperbolically convex connected fundamental region, called Dirichlet region, denoted by $D_p$. Also, $\partial D_p$ is contained in the geodesic $\{z\in \Bbb H^2:d(z,p)=d(z,\gamma p)\}$ for any $\gamma\in \Gamma\backslash \{\text{Id}\}$, i.e., hyperbolic area of $\partial D_p$ is zero.
Note that $D_p$ is closed in $\Bbb H^2$ but may not be in $\widetilde{\Bbb H^2}:=\Bbb H^2\cup \Bbb R\cup\{\infty\}$. Let $\widetilde{D_p}$ be the closure w.r.t. $\widetilde{\Bbb H^2}$. Then, $\widetilde{D_p}\backslash D_p$ may have uncountably many components. A component having positive $($Euclidean$)$length is called free side of $D_p$$($there are at most countably many free sides$)$. Any other component is a point: a point in $\widetilde{D_p}\backslash D_p$ is a proper vertex if it is the end-point of two sides$($in $\Bbb H^2)$ of $D_p$, and a point in $\widetilde{D_p}\backslash D_p$ is a improper vertex if it is the end-point of a side$($in $\Bbb H^2)$ and a free-side of $D_p$.
$\textbf{Question 1:}$ Is it possible that there is a point in $\widetilde{D_p}\backslash D_p$ that is neither proper nor improper vertex?
$\textbf{Question 2:}$ Are the number of free sides, proper or improper vertices independent of the chosen point $p$? In other words, is it possible to find some invariants for $\Gamma$ for counting free sides, proper or improper vertices $($after classifying under some equivalence relations$)$?
The above two questions are motivated by the following statements for any $p$ not fixed by any non-identity element of $\Gamma$:
$(1)$ $D_p$ has no free side $\iff$ $(2)$ $D_p$ has a finite hyperbolic area $\implies$ $(3)$ $D_p$ has finitely many sides$($in $\Bbb H^2)$ $\impliedby$ $(4)$ $\Gamma$ is finitely generated $\iff$ $(5)$ $\Bbb H^2/\Gamma$ is topologically finite.
Here, $\Bbb H^2/\Gamma$ topologically finite means $\Bbb H^2/\Gamma$ is an orbifold with finite genus, finite number of marked points$($the number of these points corresponds to the number of conjugacy classes of maximal elliptic cyclic subgroups$)$, a finite number of punctures$($the number of these punctures corresponds to the number of conjugacy classes of maximal parabolic subgroups$)$.
The correct implications are:
(0) The limit set $\Lambda$ of $\Gamma$ is the entire circle $S^1$ (i.e. $\Gamma$ is of the first kind) $\iff$ (1) $D_p$ has no free side $\Leftarrow$ (2) $D_p$ has a finite hyperbolic area $\implies$ (3) $D_p$ has finitely many sides $\iff$ (4) $\Gamma$ is finitely generated $\iff$ (5) the orbifold $\Bbb H^2/\Gamma$ is topologically finite.
The equivalence (0)$\iff$(1) comes from the fact that $D_p$ is a fundamental domain of $\Gamma$ in ${\mathbb H}^2$ and, moreover, the complement $\overline{D_p} \setminus \Lambda$ is a fundamental domain for the action of $\Gamma$ on $\overline{{\mathbb H}^2}\setminus \Lambda$.
There are many examples where $\Gamma$ is of the first kind but $D_p$ has infinite area (equivalently, $\Bbb H^2/\Gamma$ has infinite area), one is given in my answer here. The implication (2)$\Rightarrow$(1) is clear. The implications
(2)$\Rightarrow$(3)$\iff$(4)$\iff$(5)
are tricky and you can find proofs in
A.F.Beardon, "Geometry of Discrete Groups", Springer Verlag, 1983.