Photon-photon collisions do happen. As Mr. Puh said, it's very uncommon. In technical terms, the scattering cross-section is very low. High-energy photons have a higher cross-section than low-energy photons, so they are much more likely to collide. The probability of a collision goes up with frequency to the 6th power, but unfortunately I can't find a quick comparison of how likely the collision is compared to particles like electrons or protons.
Trying to deflect photons via collision runs into Heisenberg's uncertainty principle. With pool balls, we can know where they are and how fast they're moving to an incredibly high degree of accuracy. We can predict the outgoing momentum very well. With light, we'll need to narrow down the width of our target area to that cross-section that I mentioned before. (I really wish I could find a number for that.) This means the light's momentum will be uncertain, and small changes in the momentum could result in a different direction for the other photon.
One could come up with a probability distribution that describes where the photons are most likely to go afterward, but it would be much less precise than if you did this with larger objects like pool balls or even two atomic nuclei.
One other small item: You say that photons have an acceleration. This is typically not true when they're passing through empty space. They only change speed when they're moving from one medium to another (like from air to water). I'm not sure why you're bringing in acceleration here; it doesn't seem to have much to do with the question.
I hope that helps!