Here is a proof following the book by Ghys and de la Harpe:
In what follows, $X$ is a $\delta$-hyperbolic geodesic metric space.
Lemma 1. Suppose that $C\in {\mathbb R}$, $x_n, y_n\in X$ are such that $d(x_n, y_n)\le C$ for all $n$. Then $(x_n)$ converges to a point $\xi$ in $\partial X$ if and only if $(y_n)$ converges to the same point $\xi$.
I will omit the proof, see if you can prove it yourself.
Lemma 2. Suppose that $g$ is an isometry $X$, $x\in X$ and the sequence $x_n=g^n(x)$ accumulates to a point $\xi\in \partial X$. Then $\xi$ is fixed by $g$.
Proof. Suppose that $n_j$ is a strictly increasing sequence of natural numbers such that the subsequence $(x_{n_j})$ converges to $\xi$. Lemma 1 implies that the sequence
$$
(x_{n_j+1})= (g^{n_j} g(x))
$$
also converges to $\xi$. But the multiplication by $g$ transforms the sequence $(x_{n_j})$ to $(x_{n_j+1})$. Hence, $g$ fixes the limit point $\xi$ of both sequences. qed
Corollary. The accumulation sets $A_\pm$ of the sequences $(g^n(x)), (g^{-n}(x))$ is fixed by $g$ pointwise. In other words, $g$ fixes the limit set $A$ of the group $G=\langle g\rangle$ pointwise.
Lemma 3. The limit set $A$ has cardinality $\le 2$.
Proof. Suppose to the contrary that $A$ contains at least three distinct points $\xi_1, \xi_2, \xi_3$. These points are ideal vertices of an ideal triangle $T$ in $X$ (formed by three complete geodesics $c_1, c_2, c_3$ asymptotic to the points $\xi_1, \xi_2, \xi_3$). Since $g$ fixes $A$ pointwise, it "almost preserves" $T$, i.e. for each side $c_i$ of $T$, for every $n\in {\mathbb Z}$, the Hausdorff distance between $g^n(c_i)$ and $c_i$ is at most $2\delta$ (or some other function of $\delta$ depending on your definition of hyperbolicity). Let $c$ denote a center of $T$, i.e. a point within distance $\le 10\delta$ (or some other function of $\delta$) from all three sides of $T$. Then for every $n\in {\mathbb Z}$, $g^n(c)$ is also a center of $T$. Since the distance between any two centers of an ideal triangle is at most, say, $100\delta$, it follows that the $G$-orbit of $c$ is bounded. But then $g$ is elliptic and cannot have any accumulation points in $\partial G$. A contradiction. qed
Lastly, you are really handicapping yourself by not reading French. If you are seriously planning to study geometric group theory, my suggestion is to make an effort learning to read mathematical French. Unlike, say, mathematical German, it is reasonably close to the mathematical English.
Here is one:
Theorem. Let $X$ be a proper geodesic $\delta$-hyperbolic space and $f_n: X\to X$ is a sequence of $L$-bilipschitz homeomorphisms such that there exists $x\in X, C\in {\mathbb R}$ satisfying $d(x, f_n(x))\le C$. Then, after extraction, the sequence
$$
f_n: \bar{X}\to \bar{X}
$$
converges in the uniform topology. Here $\bar{X}=X\cup \partial X$ equipped with the usual topology (and the uniform structure).
A proof is an application of the Morse Lemma and chasing through the definition of the uniform structure on $\bar{X}$. One can prove a similar result by weakening $L$-bilipschitz to $(L,A)$-quasi-isometric, but that requires modifying the notion of convergence on $X$.
Best Answer
Gromov in his original 1987 book (Section 3.1) wrote a classification for arbitrary isometric group actions on hyperbolic spaces (with no further assumption) into 5 main classes. It goes at follows (the terminology is borrowed from here)
1: bounded: orbits are bounded
2: horocyclic: orbits are unbounded, $G$ acts with no hyperbolic isometry (hence there's a unique fixed boundary point)
3: lineal: there's a hyperbolic isometry and a fixed pair at infinity
4: focal: there's a hyperbolic isometry and a unique fixed point at infinity
5: general type: there's a hyperbolic isometry and no fixed point or pair at infinity.
According to Gromov, 1,2,3 are the elementary actions, i.e. the closure of an orbit in the boundary has at most 2 points, while in the non-elementary cases 4,5, this closure is uncountable.
If the action is cobounded, then case 2 (horocyclic action) can't occur. For the even more specific case of a discrete hyperbolic finitely generated group, case 4 (focal action) can't occur as well and Cases 1,3 are the virtually cyclic groups (finite and infinite).
On the other hand, all 5 cases occur for a group actions on trees, or on the hyperbolic plane (exercise).