In an X-ray tube, bremsstrahlung is generated by accelerating electrons toward a target metal. As a photon is emitted, something loses energy. I suppose that the target metal remains fixed and is re-used -- does this mean that the electron loses kinetic energy? Could it lose all of its energy and start to orbit the target atom?
Yes, the Bremsstrahlung photon's energy is taken from a diminishing electron's kinetic energy. Because the typical energies in Brehmsstrahlung are at least of order keV, it's unlikely that the process of deflection slows down the electron exactly for the electron to sit in a bound state but of course that some electrons do and electrons interacting with matter ultimately slow down and orbit the atoms (old name) to become ions (new name, after the absorption of the electron).