STM analysis reveals diverse nonequilibrium island structures formed by deposition of Ni and Al on NiAl(110) at around 300 K. Epitaxial growth in this complex alloy system is described by multisite lattice-gas modeling incorporating DFT energetics for adatoms both at adsorption sites and transition states. This approach accounts for multiple adsorption sites and diffusion paths, and accurately describes diffusion and detachment kinetics for a vast number of step-edge configurations. This is key for realistic description of island growth shapes, structure, and partial alloy ordering.