Do photons interact or not directly? All I am asking about is photon (EM wavepacket) photon (EM wavepacket) interaction.
I have read this question:
If photon-photon interactions are impossible, how are higher harmonics generated?
where Danielshank says:


As far as we know photons do not directly interact with each other.
    Yet another way to say this is that if a photon is moving along, the existence of a second photon has absolutely no influence over the first photon's path.


Photon-Photon Interactions
where JEB says:


Regarding the photons zinging around in front of you: as particles, they do interact in photon-photon scattering with a negligible cross section in the visible. As electromagnetic waves, their field strengths add linearly, which is the mechanism for con/de-structive interference; however, that only occurs when the waves are coherent. These waves are incoherent and "pass though" one-and-other.


Why do two-photon interactions only occur at extremely high energies?
So one of them says they do not directly interact, the other one says they do interact. This is a contradiction.
What I do not understand mostly, is that photons do not interact directly, two crossing photons will not scatter off (only on the second order).
Question:


*

*Do photons interact or not directly?

 A: At "tree level," the photon interacts with particles that have electric charge.  Since photons don't themselves carry electric charge, there aren't first-order photon-photon interactions.
At higher energies, or in more precise measurements, higher-order corrections make this simple statement less accurate.  Some fraction of the wavefunction for a free photon includes virtual particle-antiparticle pairs (a phenomenon known as vacuum polarization), and those virtual charged particles may interact with other photons in the electromagnetic field.  This is the lowest-order contribution to photon-photon scattering.
A: According to non-relativistic QM and classical EM alone, there should not be any photon-photon scattering.  The reason for this is that neither non-relativistic QM nor classical EM provide any cause-and-effect mechanism by which photons would scatter off of other photons.  Simply put, the photons do not have any charge, therefore no coulombic repulsion or attraction against other photons which do not have any charge either.  With regard to electron-photon scattering, at least the electron can provide some charge that leads to scattering in some way.
The quote that you mention above that seems to indicate that photon-photon scattering is impossible is ignoring additional information provided by Quantum Electrodynamics.
Quantum Electrodynamics provides some rather obscure ways that photon-photon scattering can occur.  I won't drone on with the highly technical details about how this process happens, as they justify an entire discussion of their own.
So to answer your question more directly, yes it is theoretically possible, although ordinary QM and classical EM theory don't provide the answer.
As far as experimental verification of photon-photon scattering, the wikipedia article here goes into a bit more detail, with links to the original articles I'm sure. 
Lastly, the traditional "bent rays of starlight during a solar eclipse" experiments that first validated the theory of General Relativity could be considered a photon-photon scattering experiment in the broadest sense possible (i.e. the scattering mechanism is gravitational).  I'm not going to dwell on this because this isn't what most people are talking about when they mention photon-photon scattering.
Thanks for your question.
A: Photon-photon interactions are possible in a medium (the interact through the medium). This is a common subject of study in AMO (atomic, molecular and optical) physics. Examples of phenomena that come from this include Electromagnetically Induced Transparency and Four-wave mixing.
In vacuum: as mentioned in a previous answer by "my2cts", vacuum polarization has been observed, but this is extreme phenomena that is observed at very high fields. As a sidenote, a lot of people are currently looking for vacuum birefringence.
A: Photons do interact via virtual electron positron pairs. 
