The acceptor (N)-isovalent (S, Se, Te) elements that separately co-dope ZnO is a novel doping technology. However, the effects of triaxial strain on the photocatalytic performance and mechanism of doped systems are often neglected in experimental studies. Moreover, the influence of interstitial H on the doped ZnO system is not considered. This paper adopted first-principles and selected the appropriate strain range and step length. Assuming the equivalent effect of several strain points of tensile strain or compressive strain, we used simulation calculation to fit the complex strain problem in the actual process (such as doping, the strain problem caused by the difference of the lattice constant and thermal expansion coefficient of the substrate and the doping system), and then studied physical properties of the doped system. The planar wave ultrasoft pseudopotential + U method based on the first-principles generalized gradient approximation under the framework of density functional theory was used to calculate (S, Se, Te) and 2 N co-doped ZnO. The interstitial H-doped Zn36SO33N2 was studied as a representative ZnO system. Results showed that the binding energy of all doping systems under unstrained conditions was negative with a relatively high stability. The strain increased under tensile strain and compressive strain. As the formation energy of the doping systems increased, system stability decreased. Although the band gap of all compressive strain systems widened, impurities formed in the band gap, and valence band electrons absorbed photons of low energy to transition to the energy level of the impurities and then to the energy level of conduction band. The photoexcited electron-graded transitions induced the absorption spectrum to red-shift. The band gap of the system widened, whereas the band gaps of other tensile strain systems narrowed, except for when the tensile strain was 1%. However, regardless of whether the band gap of the tensile strain system became narrow or wide, the photoexcited electrons were still able to transition in stages. Hence, the absorption spectrum of the tensile strain system still red-shifted. The stability, activity, carrier separation, and lifetime of Zn36SHiO33N2 were relatively better than those of Zn36SO33N2 at 5% tensile strain. These properties make Zn36SHiO33N2 an ideal photocatalyst. This study provides a theoretical reference for the experimental design and preparation of novel ZnO photocatalysts.