We use numerical linked-cluster expansions to study finite-temperature properties of strongly interacting fermions in two-dimensional optical lattices, governed by the Hubbard model. We show the double occupancy and entropy for the infinite homogeneous system at temperatures significantly lower than those obtained by other exact methods at strong interactions. Employing a local density approximation, and using the high-precision results for the entropy, we study the density and nearest-neighbor spin correlation profiles of lattice fermions trapped in a harmonic potential during adiabatic processes. Starting with a trap that has a substantial band-insulator region at high temperatures, we show how one can access the Mott region at low temperatures by flattening the trapping potential.