> 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 topology, to make it a metric space. 

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"?

Surely this is discussed in the reference you are reading.  If you give a link to it, perhaps we can find the location of this result. 

This is also discussed in Thurston's notes.