What are the properties of a photon with Planck energy? Is it even possible to interact with it, or does it just travel trough all matter?
The Planck energy is an approximate energy, not an exact one. It is believed to be the energy range at which the quantum effects of gravity become important. They talk a lot about the Planck energy on TV. However our best current theory, the Standard Model, is not expected to remain valid at such a high energy, and what actually does happen at the Planck energy, to the photon or to any other particle, is at this stage pure speculation.
To reach the Planck scale, you have to collide two particles with an energy of the order of the Planck scale. A photon with the Planck energy (in our lab), colliding with something on earth would be certainly a very interesting collision, but it would not go beyond the limit of our current theories (there could be new physics, but there does not have to be). The energy of a photon is frame-dependent. For every photon, there is a frame where its energy reaches (or even exceeds) the Planck scale.
Let's collide the photon with a proton: (E_P,E_P,0,0) + (m_p,0,0,0) leads to a center of mass energy of ##\sqrt{(E_P+m_p)^2-E_P^2} \approx 5 EeV = 5 \cdot 10^6 TeV##. More than we can produce in collider experiments, but way below the Planck scale of 10^{16} TeV where gravity becomes significant.
There is a Greisen–Zatsepin–Kuzmin limit for cosmic radiation - about 8 joules. This is far less than the 2 billion joules for your photon. So, if this particle could exist at all, it would immediately begin breaking apart into less energetic particles because of its interaction with the microwave background radiation. But before that happened, I would wonder if it would constitute a tiny black hole that would instantly evaporate - releasing a shower of other particles.
I don't think he meant to. The GZK limit applies to high-energy protons, not photons. (But we should still be somewhat skeptical about any extrapolation of current theory to such a remarkably energetic photon).
Photons can indeed interact indirectly with other photons. This is a purely quantum mechanical effect that is basically unavoidable as long as the photon interacts with any other charged particles. One will always have processes where there is a virtual charged particle in a loop of the diagram for a process, with some number of external photons attached to the loop. It turns out that at least 4 photons must be involved (see this thread for a discussion of the kinematical reasons why), so, for example, a pair of photons in the initial state can interact in such a way that there is still a pair of photons in the final state, which is what we'd usually think of as a light-by-light scattering process. Edit: I should add that the amplitude for the lowest-order process for light-by-light scattering, at high enough energies, is about 10000 times smaller than the amplitude for photon-electron scattering.