Relativity and photon interactions A particle's interaction (with anything it can interact with) can be thought of as it making a measurement of the physical quantity associated with the interaction, (e.g. electric field in case of the interaction  between charged particles) and acting accordingly. To make a measurement one first needs a frame to make measurements "in" (a Lorentz frame in light of relativity). Assigning such frame to a photon appears to me, to be problematic in the sense that constant velocity of a photon in any inertial frame implies a photon in its own frame having a velocity $c$ (the speed of light). Now if we assume that no such frame exists for a photon, photons cannot interact with an other one. 
Can we explain in this way that photons do not interact with each other, or more generally particles moving at velocity $c$ do not interact with similar particles?
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A particle's interaction (with anything it can interact with) can be thought of as it making a measurement of the physical quantity associated with the interaction, (e.g. electric field in case of the interaction between charged particles) and acting accordingly.

When we use the term "particle" with "interaction" we are talking of elementary particles and we are in the framework of quantum mechanics. To make a measurement, yes,it is correct that an interaction between elementary particles has to take place. This is the table of elementary particles, and at the bottom line it is interactions between them that form the world we observe in the microworld, and collectively the world we observe in the macroworld we live and move in.  The macro world emerges from innumerable interactions of the micro world constituents.
A field is not an elementary particle. A field in second quantization manifests elementary particles according the  operators operating on the field, so it is not as simple. To measure the electric field a huge number of photons are involved and it is not a simple example as you think.

To make a measurement one first needs a frame to make measurements "in" (a Lorentz frame in light of relativity). 

A lorenz frame can be any frame correctly defined. All interactions observed in that frame can be transformed to other moving frames, but we tend to work with lorenz invariant quantities so as not to worry about transformations,

Assigning such frame to a photon appears to me, to be problematic in the sense that constant velocity of a photon in any inertial frame implies a photon in its own frame having a velocity c (the speed of light). 

A photon has no rest frame, is what you mean. There is no reason for it to have a rest frame other than the prejudices we carry from the macro world. The lorenz transformation assures that the mathematics of any frame are correct for photons too.

Now if we assume that no such frame exists for a photon, photons cannot interact with an other one.

Two photons define a rest frame , because two photons have an invariant mass which has a frame where all momenta are zero. Example the pi0 decay to two photons.

Can we explain in this way that photons do not interact with each other, or more generally particles moving at velocity c do not interact with similar particles?

No, we cannot, because they do interact through exchanges of virtual particles at higher orders in the mathematical expansions of the solutions of the specific problem, and thus with small probabilities. Gamma gamma scattering experiments exist, and gammas are high energy photons. They are even talking of gamma gamma colliders,
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A particle's interaction (with anything it can interact with) can be thought of as it making a measurement of the physical quantity associated with the interaction,

This isn't true. All measurements are interactions, but not all interactions are measurements.
When we talk about a Lorentz frame in special relativity, we're talking about a very sophisticated construct, which is sort of like filling the universe with clocks, rulers, and signaling devices, and then carrying out a complicated surveying process. A single photon doesn't do anything like this.
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A particle's interaction(with anything it can interact with) can be thought of as ,it making a measurement (of the physical quantity associated with the interaction-eg electric field in case of charged particles interaction)and acting accordingly

This isn't the correct way to think about interactions, and as you later point out this (kind of) violates causality. When two particles make a measurement of each other they have already interacted!
Firstly, you should expand the framework that you're trying to solve this problem in. In order to fully describe particle interactions you have to move from describing individual relativistic particles to talking about relativistic interacting fields whose quanta are particles, or a 'quantum field theory'. The reason for this is that the photon itself is a quantum of the the electromagnetic field which itself is fundamentally relativistic. In this framework, interacting fields are constantly interacting, so your problem does not arise.
