We study the effect of polymer coating, pressure, temperature and light on the electrical characteristics of monolayer WSe₂ back-gated transistors with quasi-ohmic Ni/Au contacts. We prove that the removal of a layer of poly(methyl methacrylate) or a decrease of the pressure change the device conductivity from p to n-type. We demonstrate a gate-tunable Schottky barrier at the contacts and measure a barrier height of ~70 meV in flat-band condition. We report and discuss a temperature-driven change in the mobility and the subthreshold slope which we use to estimate the trap density at the WSe₂/SiO₂ interface. We study the spectral photoresponse of the device, that can be used as a photodetector with a responsivity of ~0.5 AW^-1 at 700 nm wavelength and 0.37 mW/cm² optical power.