We report on a detailed study of terahertz (THz) photoresistivity in a strained HgTe three-dimensional topological insulator (3D TI) for all Fermi level positions: Inside the conduction and valence bands and in the bulk gap. In the presence of a magnetic field, we detect a resonance corresponding to the cyclotron resonance (CR) in the top surface Dirac fermions (DFs) and examine the nontrivial dependence of the surface state cyclotron mass on the Fermi level position. We also detect additional resonant features at moderate electron densities and demonstrate that they are caused by the mixing of surface DFs and bulk electrons. At high electron densities, we observe THz radiation-induced 1/B-periodic low-field magneto-oscillations coupled to harmonics of the CR and demonstrate that they have a common origin with microwave-induced resistance oscillations previously observed in high mobility GaAs-based heterostructures. This observation attests the superior quality of a 2D electron system formed by helical surface states in strained HgTe films.