When an electron emits a photon, which direction does it go?

Even though they have no m, photons have momentum, which must be conserved. So a photon being re-emitted will either be ped along with its original momentum for a transparent medium, or bounced back like a ball off a wall for a reflective medium. Photons due to energetic processes in the material (heat, for example) are emitted as spherical wavefronts, constrained by directionality imposed by the geometry of the emitter. So a photon emitted from a flat surface will have a wavefront that covers a hemisphere. An individual photon will interact with matter at only a single point, of course, but the wavefront shows the probability distribution of that interaction.