In part I of this series, we showed that the observed Universe can be modeled as a local Black Hole of fixed mass M≃6×1022M⊙, without Dark Energy: cosmic acceleration is caused by the Black Hole event horizon rS = 2GM. Here, we propose that such Black Hole Universe (together with smaller primordial Black Holes) could form from the hierarchical free-fall collapse of regular matter. We argue that the singularity could be avoided with a Big Bounce explosion, which results from neutron degeneracy pressure (Pauli exclusion principle). This happens at GeV energies, like in core collapse supernova, well before the collapse reaches Planck energies (1019 GeV). If our Universe formed this way, there is no need for Cosmic Inflation or a singular start (the Big Bang). Nucleosynthesis and recombination follow a hot expansion, as in the standard model, but cosmological measurements (which are free parameters in the standard model) could in principle be predicted from first principles. Part or all of the Dark Matter could be made up of primordial compact objects (Black Holes and Neutron Stars), remnants of the collapse and bounce. This can provide a faster start for galaxy formation. We present a simple prediction to explain the observed value of M≃6×1022M⊙ or equivalently ΩΛ (the fraction of the critical energy density observed today in form of Dark Energy) and the coincidence problem Ωm∼ΩΛ.