A new formulation of the polymer reference interaction site model (PRISM) theory for suspensions of hard spheres and rigid rods has been developed. The nonlocal loss of orientational entropy when a rod is near an impenetrable particle is accounted for in a thermodynamically self-consistent manner. In the ideal needle limit, quantitative predictions and qualitative scaling behaviors are determined for the depletion potential, rod insertion chemical potential, intermolecular pair correlations, and fluid-fluid spinodal phase- separation boundaries as a function of rod-particle size-asymmetry and mixture variables. The needle-induced depletion potential of mean force between a pair of particles is in good agreement with exact numerical results and experiments on dilute silica-virus suspensions. Needle-induced clustering of colloidal particles is studied, and the PRISM results are in good agreement with the small amount of available simulation data. The behavior of spinodal boundaries in the extreme size-asymmetry ratio limits is predicted to be qualitatively different, in contrast with that reported from prior approaches based on free volume/scaled particle theory or a density functional approach that finds that a universal description applies. The influence of rod-rod repulsions far from the isotropic-nematic phase transition is found to be minor.