We use self-consistent periodic Density Functional Theory to study GaN surface cleaning, contamination, passivation and adsorption of small organic and inorganic molecules on GaN(0001) in terms of surface chemistry and electronic structure. Coverage-dependent binding energies of H, O, C, N, OH, H₂O, Cl, NO, NH, NH₂, NH₃, as well as a few short alkanes and alkenes on the GaN(0001) surface have been calculated. Adsorbed H and Cl species on GaN(0001) shift available states in the GaN band gap towards the conduction band, thereby passivating the surface. Chemical surface passivation after aqueous HCl surface treatment is confirmed by XPS experiments. Furthermore, we developed a full potential energy surface (PES) for NH₃ and H₂O decomposition. The PES shows that NH₃ readily decomposes to NH and H species, but the formation of atomic N by N-H bond scission is endothermic and hindered by a large activation energy barrier of 3.4 eV. Water almost spontaneously dissociates into OH and H upon adsorption on GaN(0001), and can further dissociate into O and H with a barrier of ca. 1.1 eV. In agreement with DFT calculations, XPS data show a considerable O/Ga ratio for NH₃ treated GaN surfaces, whereas the O/Ga ratio decreases significantly after HCl treatment. Water formation from surface adsorbed atomic O and H species is endothermic (2.6 eV) which may reflect some difficulty for the removal of oxygen contaminants on GaN(0001) through a NH₃ treatment. We also studied the reactive attachment of C₂H₄ to the hydrogen covered GaN(0001) surface at 1/4 ML and 3/4 ML initial H coverage. The results clearly indicate how higher H coverages thermodynamically and kinetically favor the formation of C₂H₅ as compared to lower H coverages.