Liquid-state integral equation methods are employed to study the thermodynamic and structural properties of ideal and repelling rigid rods mixed with hard spheres in the limits when one of the species is dilute. The role of rod aspect ratio and sphere/rod size asymmetry is explored over a wide range of system parameters encompassing the colloid, nanoparticle, and crossover regimes. Novel predictions are found for the polymer (sphere) mediated depletion potentials and second virial coefficients of particles (rods) in dense polymer (sphere) suspensions. The adequacy of the closure approximations employed is tested by comparison with available numerical calculations and more rigorous theories in special limits. The liquid-state theory appears to be accurate for all properties in the nanoparticle regime and for the insertion chemical potential of needles and spherocylinders. However, it significantly underestimates depletion attractions effects in the colloidal regime of short rods and large spheres due to nonlocal entropic repulsion effects between polymers and particles not captured by the classic Percus-Yevick approximation.