Photons carry no charge, so normally they don't interact with each other. However, at high enough energies they do. The explanation given by the best available theory is that there is enough energy that a virtual electron and virtual anti-electron form, and since those have charges, they interact, and from the outside it appears that the photons interacted.
This is totally impossible in the classical view, but is fine in the quantum electrodynamic (QED) view.
High energies are not absolutely necessary. Couples of "virtual" particles can be created at any energy, it's just less likely for smaller energies.
I think the real problem was the experimental set-up: particle accelerators use electric field to transfer energy to particles and magnetic field to control their trajectory and eventually to cause collisions, but all of this only works with charged particles. And photons carry no charge, as you said!
As the article mentions, the observation was finally made possible by accelerating bunches of lead ions, as they carry around a cloud of high energy photons (I suppose due to bremsstrahlung caused by chaotic movements within the bunches).
No, it doesn't. Remember that the mass of a particle is just an average: energy and time are linked by the uncertainty principle as position and momentum are, thus decaying particles (pretty much all of them) do not have a sharply defined mass.
No. For example the weak force is created by the very heavy W and Z bosons.
The uncertainty principle says you can "fudge" the energy if your time-scale is short enough, and this is what happens. A result of this is that the distance the weak force can operate over is limited, since the timescale has to be short, so the particles don't have the time to travel very far.
Virtual means that the particle does not obey the Einstein energy-momentum equation: E^2 = p^2c^2 + m^2c^4 . You might recognize the shorter form, where the particle has zero momentum, as E=mc^2.
>Photons carry no charge, so normally they don't interact with each other.
Not having charge doesn't imply they don't interact. They just don't through the electromagnetic force. Neutrons also have no charge but interact via the strong force, for example.
They think they got photons to bounce off each other. It's a really old prediction but really rare, so hard to prove. They think it happened 13 times across 4 billion events.
This is totally impossible in the classical view, but is fine in the quantum electrodynamic (QED) view.