Using an effective logarithmic-Gaussian pair potential that models the interaction between star polymers, we compare the hypernetted chain (HNC) and random phase approximations (RPA) for calculating the bulk structure (including the Fisher-Widom and Lifshitz lines), thermodynamic functions and phase diagram of a phase-separating binary fluid of star polymers, of two-arm length ratio 2:1. Thereby, the stars considered here are equivalent to linear chains in the mid-point representation of their effective interaction. We find that at densities where the star coronas overlap, the quasi-exact HNC and RPA give very similar results. Using a density functional approach, with a functional which generates the RPA, we calculate properties of the inhomogeneous binary fluid. We determine the surface tension and one-body density profiles at the free fluid-fluid interface. For states well removed from the critical point the profiles exhibit pronounced oscillations. For a purely repulsive planar wall potential that models the effective potential between a star polymer and a hard wall, we find a first-order wetting transition with the associated pre-wetting line.