1) Can you point me out the rigorous definition of "limit of a sequence of arcs" on a surface?

2) In which topological space are we considering the limit of arcs to obtain a lamination?

Suppose that $S$ is a orientable surface of finite topological type, equipped with a hyperbolic metric of finite volume. Consider the set $A$ of simple geodesic arcs properly embedded in $S$. Consider also the larger set $C$ of closed subsets of $S$. We equip $C$ with the Hausdorff metric.

Suppose that $a_n \in A$ is a sequence of arcs. This also gives a sequence in $C$. If $a_n$ has a limit in $C$, then that is the desired object. It is a theorem that the limit is geodesic lamination: a closed subset of $S$ that can be realized as a disjoint union of simple geodesics.

It is an easy exercise to find a geodesic lamination that is *not* a Hausdorff limit of arcs.

3) Where can I find a proof of the statement "every sequence of arcs converges to a lamination on a hyperbolic surface"?

It is not true that *every* sequence converges. Here is the correct statement, from Section 8.5 of Thurston's notes: "An easy diagonal argument shows that every sequence $\{\gamma_i\}$ has a subsequence which converges geometrically." More detail is given in the proof of Proposition I.4.1.7 of Notes on notes of Thurston by Canary, Epstein, and Green.