The sub-Chandrasekhar-mass double-detonation (DDet) scenario is a contemporary model for Type Ia supernovae (SNe Ia). The donor star in the DDet scenario is expected to survive the explosion and to be ejected at the high orbital velocity of a compact binary system. For the first time, we consistently perform 3D hydrodynamical simulations of the interaction of supernova ejecta with a helium (He) star companion within the DDet scenario. We map the outcomes of 3D impact simulations into 1D stellar evolution codes and follow the long-term evolution of the surviving He-star companions. Our main goal is to provide the post-impact observable signatures of surviving He-star companions of DDet SNe Ia, which will support the search for such companions in future observations. Such surviving companions are ejected with high velocities of up to about 930 km s(-1). We find that our surviving He-star companions become significantly overluminous for about 10(6) yr during the thermal re-equilibration phase. After the star re-establishes thermal equilibrium, its observational properties are not sensitive to the details of the ejecta-donor interaction. We apply our results to the hypervelocity star US 708, which is one of the fastest unbound stars in our Galaxy; it travels with a velocity of about 1200 km s(-1), making it a natural candidate for an ejected donor remnant of a DDet SN Ia. We find that a He-star donor with an initial mass of greater than or similar to 0.5 M-circle dot is needed to explain the observed properties of US 708. Based on our detailed binary evolution calculations, however, a progenitor system with such a massive He-star donor cannot get close enough at the moment of the SN explosion to explain the high velocity of US 708. Instead, if US 708 is indeed the surviving He-star donor of a DDet SN Ia, it would require the entire pre-supernova progenitor binary to travel at a velocity of about 400 km s(-1). It could, for example, have been ejected from a globular cluster in the direction of the current motion of the surviving donor star.