Since the electron and photon are quantum mechanical objects, the angle at which the electron and photon move after the collision is probabilistic. As you have pointed out, they do not actually "collide" but instead they interact. You can think of this process as the electron absorbing the photon at one point, and then re-emitting one at another.
The scattering angle is therefore random, but the interaction does conserve energy and momentum. At the "instant" the interaction occurs, the scattered photon and electron are in a superposition of all possible states and all these possible states form a probability distribution, and this probability distribution can be computed using the appropriate quantum mechanical techniques, like QED and Feynman diagrams.
what is it about the physical conditions before the collision that determines what direction these particles move after the collision?
There are no exact specific conditions before the interaction that will tell you with certainty what direction the particles will travel. You are sort of thinking classically. Again, all quantum interactions are probabilistic. Even though we may calculate the differential cross-section as a function of angle (see the Klein-Nishina formula), the actual angle the particles take will be determined with certainty only after the measurement (collapsing the wave functions).