Question 1:

in $\mathrm{Isom}(\mathbf{H}^n)$, the centralizer of any loxodromic element preserves its axis, and hence is contained in a closed subgroup isomorphic to $\mathrm{O}(n-1)\times\mathrm{Isom}(\mathbf{R})$. In particular, discrete subgroups of the latter are virtually cyclic.

In addition, if a subgroup has no loxodromic element, either it has compact closure, or is horocyclic, in the sense that it fixes a unique point at infinity and is contained in the leafwise stabilizer of the corresponding horosphere foliation with respect to some point at infinity, which is isomorphic to $\mathrm{Isom}(\mathbf{R}^{n-1})$. The centralizer of a horocyclic subgroup fixes the same point at infinity, hence, in turn, is either relatively compact (pointwise fixing an axis), horocyclic, axial, or focal in the sense that it preserve (maybe not leafwise) the corresponding foliation. As regards discrete subgroups, they can't be focal.

Hence, if we have two discrete subgroups $G_1,G_2$ of $\mathrm{Isom}(\mathbf{H}^n)$ centralizing each other, we have one of the following, showing that there are fery few subgroups decomposing as direct products.

- both are virtually cyclic (each with a loxodromic element)
- one is virtually cyclic (with a loxodromic element) and the other one is virtually abelian (being horocyclic)
- both are virtually abelian (horocyclic)
- one is finite (and the other has to preserve its subspace of fixed points). In particular, if the other one does not preserve any proper geodesic subspace (e.g., is Zariski-dense), then it has a trivial centralizer.

As I said in a comment, Question 2 is too naive: just consider a subgroup of order 6. But also there are torsion-free subgroups decomposing as nontrivial direct products, for $n\ge 3$, such as a horocyclic copy of $\mathbf{Z}^2$. Geometrically this correspond to a 3-dimensional cusp, homeomorphic to $\mathbf{R}\times\mathbf{T}^2$, so well, this is indeed homeomorphic to a product of two non-contractible manifolds. By the previous description, this is essentially the only possibility, namely every real hyperbolic manifold that homotopically decomposes as a product has a finite covering homeomorphic to the product of a Euclidean space and a torus.