The probability for a positron (e+) incident upon a metal surface either to be converted into a positronium atom (Ps) which leaves the surface or to be reemitted as a bare e+ is calculated using a model which goes considerably beyond that used in a previous calculation. The Ps conversion and e+ emission process proceeds in three stages: implantation, thermal diffusion from the characteristic stopping distance to the surface, and electron capture with bare e+ escape. The stopping distance and its rms deviation are calculated now including both the effect of elastic positron ion-core scattering (in addition to inelastic e+ conduction-electron and e+-phonon scattering) and finite temperature corrections in the e+-phonon scattering. Given a thermal e+ near the surface, we examine the probabilities of the following processes: direct e+Ps conversion (calculated within a simple "resonant-level" model), e+ trapping into the image potential induced surface state, direct e+ escape into vacuum (perturbation theory), and Ps formation via e+ detrapping from the surface state. Overall conversion and emission efficiencies are calculated by solving an appropriate diffusion equation with boundary conditions set by the branching ratios for the above surface processes. The temperature and incident energy dependence of our results for final Ps conversion and e+ emission efficiency are compared with experiment.