The roughening behavior of moving surface under a deposition and evaporation dynamics is explored within the hypercube-stacking model. One limiting case of the model is an equilibrium surface, which exhibits thermal roughening for surface dimension d≤2. Another limiting case is nonequilibrium irreversible growth, where the model is shown to map exactly to zero-temperature directed polymers on a hypercubic lattice with a random energy distribution. Results of exact calculations for d=1 and of large-scale Monte Carlo simulations [N=220, 11 5202, and 2×1923 surface sites for d=1, 2, and 3, respectively] are presented that establish the Kardar-Parisi-Zhang equation as the correct continuum description of the growth process. For pure deposition (i.e., irreversible growth), careful analysis of surface width data yields the exponents β(2)=0.240±0.001 and β(3)=0.180±0.005, which violate a number of recent conjectures. By allowing for evaporation, we observe a less rapid increase of the surface roughness as a function of time. This phenomenon is consistently explained by a crossover scenario for d=1 and 2 but a nonequilibrium roughening transition for d=3, as predicted by a perturbative renormalization-group analysis of the Kardar-Parisi-Zhang equation. Detailed predictions on crossover scaling from the renormalization-group analysis are also confirmed by simulation data. In the d=1 case, some of the continuum parameters characterizing the renormalization-group flow are obtained explicitly in terms of the lattice parameters via the exact calculation of steady-state properties of the model.