Do photons annihilate each other? If yes, what do they produce? If no why not? If photons are their own anti-particles, why don't they annihilate? 
Also on a side note, why do particles and anti-particles annihilate?
 A: A real particle-antiparticle pair can annihilate and create two real photons. Quantum electrodynamics is time-reversal invariant, so the opposite process, in which two real photons disappear and create a real particle-antiparticle pair, is also possible.
But this isn’t called “photon annihilation”; it’s called “two-photon pair production”. “Annihilation” is usually reserved to mean the transformation of a massive particle-antiparticle pair into massless particles.
The simplest answer to “Why do particles and antiparticles annihilate?” is “Because they can!” A slightly more complicated answer is that QED Feynman diagrams have only one kind of “vertex”, and it represents six different things depending on which particles (either real or virtual) are incoming and which outgoing: a charged particle absorbing a photon; a charged particle emitting a photon; a charged antiparticle absorbing a photon; a charged antiparticle emitting a photon; a charged particle and its antiparticle disappearing and creating a photon; a photon disappearing and creating a charged particle and its antiparticle. In quantum field theory these are all fundamentally the same interaction.
A: 
Do photons annihilate each other? If yes, what do they produce? If no why not?

In the standerd model of particle physics there  is no photon-photon vertex, as for example there is an  electron-positron vertex. Electrons and positrons annihilate into many particles because the quantum numbers of electron and positron add up to zero, thus  just energy and momentum can be left to go into any other pairs  with  quantum number that add up to zero .
Photons have just their four vector and as there is no direct vertex have to go through a virtual charged particle loop to be able to scatter off each other ,


A Feynman diagram (box diagram) for photon–photon scattering, one photon scatters from the transient vacuum charge fluctuations of the other

When the energy is high enough instead of photons coming out on the right one could have particles as in an e+e- collider, and there are studies to make gamma  gamma colliders.
In that sense the added two incoming photon four vector which carries zero quantum numbers,  will become other particles, their quantum numbers adding up to zero, equivalent to an annihilation. At low photon energies there is not enough energy to create particle pairs and so the different terminology.
