Neutron-rich nuclei around 78Ni provide an important testing ground for the $N=50$ shell closure and cross-shell correlations. We develop a nuclear-structure implementation of the semistochastic heat-bath configuration interaction (SHCI) method and apply it, for the first time, to nuclear structure calculations with ab initio valence-space Hamiltonians. SHCI builds a compact variational space through heat-bath selection and accounts for remaining correlations using semistochastic second-order perturbation theory. Combined with valence-space in-medium similarity renormalization group Hamiltonians derived from chiral two- and three-nucleon interactions, SHCI provides a large-space solver for extended proton- pf and neutron- gds valence spaces beyond conventional exact diagonalization. Benchmark calculations for 78Ni show that SHCI achieves faster convergence of low-lying spectra than comparable particle-hole truncations, demonstrating the efficiency of determinant-based selection. We further study the $N=50$ isotones and the Ni, Cu, and Zn isotopic chains toward the predicted neutron-rich limits. The results indicate a robust but gradually weakened $N=50$ shell closure from Ni to Zn, predict the drip-line nuclei, and suggest a possible weak subshell structure near $N=58$. This work establishes SHCI as a systematically improvable large-space solver for ab initio nuclear structure.