We report density functional calculations of the atomic and electronic structure of the spinel phases ZnRh2O4 and ZnIr2O4 as well as crystalline ZnO lightly doped (1 at.%) with Rh and Ir ions using the B3LYP hybrid functional. Calculations for the spinels show band gaps (∼3 eV) and lattice parameters (∼2% difference) in reasonable agreement with experimental data. Incorporation of the transition metals into ZnO induces local distortions in the lattice and the appearance of metal d levels in the low gap region and near the conduction band minimum, with a d-d splitting larger than 2 eV, which helps maintain transparency in the material. Addition of a hole to the simulation cell of both spinels and doped ZnO leads to charge localization in the neighbourhood of Rh/Ir accompanied by local lattice deformations to form a small polaron which may lead to low hole mobility. We calculate polaron diffusion barriers in the spinels and obtain values around 0.02-0.03 eV. These very low barrier energies suggest that at high Rh/Ir concentration hole conduction occurs mainly by the band conduction mechanism at room temperature. We also develop models of the amorphous spinels by means of classical molecular dynamics simulations, and observe a marked reduction in the coordination number of Rh/Ir, from 6 to 4, in the amorphous phase, which may reduce transparency in these materials.