This is an improved version of this question (sorry if I wasn't so clear there and sorry if this is well-known and I didn't find the reference).
Let $N$ be a hyperbolic 3-manifold of finite volume and let $N_1,\,N_2,\,\ldots,\,N_k$ be pairwise disjoint representatives of the cusp ends of $N$, so each $N_i$ is diffeomorphic to $\mathbb{T}^2\times [0,\infty)$. Let $M = N\setminus (N_1\cup\ldots\cup N_k)$. Then, $M$ is a compact 3-manifold with incompressible toroidal boundary.
1) After performing Dehn filling on the cusp ends of $N$ so that the resulting closed manifold $\widehat{N}$ is hyperbolic, is there link $\Gamma\subset \widehat{N}$ such that $N$ is diffeomorphic to $\widehat{N}\setminus \Gamma$?
2) Assuming 1) is true, how different Dehn fillings on the above procedure change $\widehat{N}$ and $\Gamma$ as above? In other words, if we produce $\widehat{N}_1$ and $\widehat{N}_2$ trough distinct Dehn fillings in $N$ are $\widehat{N}_1$ and $\widehat{N}_2$ diffeomorphic? If they are, are the links $\Gamma_1$ and $\Gamma_2$ isotopic?
I am trying to understand (say orientable) hyperbolic 3-manifolds (of finite volume) and the standard examples are link complements in a closed three manifold, so this question is more or less asking if for a given hyperbolic 3-manifold of finite volume there is only one way of seeing it topologically as a link complement. Thank you.