Let Ω be a smooth relatively compact domain containing zero in the Poincaré ball model of the Hyperbolic space Bn (n≥3) and let −ΔBn be the Laplace-Beltrami operator on Bn. We consider problems of non-existence, existence, and multiplicity of variational solutions for the borderline Dirichlet problem,(1){−ΔBnu−γV2u−λu=V2⋆(s)|u|2⋆(s)−2u in Ωu=0 on ∂Ω, where γ<(n−2)24, 0<s<2, 2⋆(s):=2(n−s)n−2 being the corresponding critical Sobolev exponent, while V2 (resp., V2⋆(s)) is a Hardy-type potential (resp., Hardy-Sobolev weight) that is invariant under hyperbolic scaling and which behaves like 1r2 (resp. 1rs) at the origin. The bulk of this paper is a sharp blow-up analysis that we perform on certain approximate solutions of (1) with bounded but arbitrary high energies. When these approximate solutions are positive, our analysis leads to improvements of results in [5] regarding positive ground state solutions for (1), as we show that they exist whenever n≥4, 0≤γ≤(n−2)24−1 and λ>0. The latter result also holds true for n≥3 and γ>(n−2)24−1 provided the domain has a positive “hyperbolic mass”. On the other hand, the same analysis yields that if γ>(n−2)24−1 and the mass is non vanishing, then there is a surprising stability of regimes where no variational positive solution exists. As for higher energy solutions to (1), we show that there are infinitely many of them provided n≥5, γ<(n−2)24−4 and λ>n−2n−4(n(n−4)4−γ).