During particle-antiparticle annihilation, are the photons expelled perfectly perpendicular to the original direction of the particle-antiparticle pair?
In the center-of-momentum frame, the two photons can come out back-to-back in various directions, with a probability distribution. Any direction is consistent with energy-momentum conservation.
When non-relativistic electrons and positrons annihilate, all directions are equally likely for the photons; the differential cross section is isotropic. When the electron and positron are ultra-relativistic, the photons strongly tend to be along the direction of motion the annihilating particles. (This is for unpolarized beams.)
The calculation of the differential cross section for any energy may be found here.