What would the collision of two photons look like? Could someone explain to me what the collision of two photons would look like? Will they behave like,


*

*Electromagnetic waves: they will interfere with each other and keep their wave nature

*Particles: they will bounce like classical balls


I assume that energy of that system is too small to make creation of pairs possible.
 A: A lowest order QED Feynman diagram for the process photon + photon $\rightarrow$ electron + positron looks like shown below (the time axis is the horizontal axis).
From the point of view of energy conservation, this process is only possible if sum of the energy of the photons is above twice the electron mass. In the center of mass frame of the di-photon system, the photons need to have at least 511 keV.

A: By time symmetry, two photons with sufficient mutual energy necessarily are capable of annihilating each other to produce an electron-positron pair, since one of the decay modes for positron-electron annihilation is the production of two gamma photons. It's just harder to arrange experimentally, since unlike the electron and positron the uncharged photons have no attraction for each other.
Here's an experimental exploration of two-photon positron production: 
D.L Burke et al, Positron Production in Multiphoton Light-by-Light Scattering, SLAC, June 1997.
Photon-photon interactions (scattering) via virtual particle pairs gives you two-photon physics, which looks at the probabilities of photon-photon production of particle pairs much heavier than electrons.
A: Photons do not have the feature of self interaction, meaning that two photons can neither attract nor repell each other. Therefore two photons can not collide.
A: Your assumption that pair production is ruled out, rules out* that two photons interact through higher-order processes. Quantum electrodynamics tells us that two photons cannot couple directly. That leaves us with classical electromagnetism, which tells us that electromagnetic waves pass through each other without any interference.
*Edit. The photons can interact through higher-order processes. As pointed out in the comments (and I hope I'm getting this right), there is a (quite small) probability amplitude for two photons to get absorbed in, and two photons be emitted by, e.g. an fermion-antifermion virtual pair (which is the leading contributor to the combined amplitude of all such processes). Whether (and this is my cop-out) the emitted photons can be considered the same photons as the absorbed photons, I leave to the, certainly more knowledgeable, commentators.
