What are the properties of light in the wave/ particle theory that map to properties we physically observe? When we see two beams of light of different colours, what is different about them in the wave theory? And what is different about the photons that make up each of them?
When we see two beams of light, one more intense than the other, what is different about them in the wave theory, and what is different about them in the photons that make them up?
When we say two sources of light are in-phase/ out of phase, how do we physically know this? And again, what is similar about two beams that are in phase in the wave model, and the particle model?
I have part-answers and muddled up answers from different sources, I think it's good to have one comprehensive answer covering this.
 A: Two colors (assuming they are the same amplitude) have different wavelengths and thus frequencies. Thus the photons have different energies.
Two intensities of light (assuming they are the same color), the wavelengths and frequencies are the same but the amplitude in the more intense wave is higher, and the photons are the same energies, but there are more photons in the more intense beam. 
When two beams of light that are in phase (assuming they are coherent and the same color), we know this because they would exhibit constructive interference, i.e. the intensities add up (not subtract, as in destructive, out of phase beams). Their wavelengths and frequencies are the same, but the amplitudes add if they are constructive, or subtract if they are destructive; and the energies are the same, but the number of photons add up if they are in phase, and subtract if out of phase, where the interference occurs.
A: Photons are particles with oscillating electric and magnetic field components. What water waves and photons have in common is the energy transfer without mass transport and the oscillation of the energy content. The difference is that for a medium the atoms and molecules vibrate but do not move in the direction of wave dissipation, while for light the photons move (at the speed of light) and during their propagation their magnetic and electric fields oscillate (for each photon its own fields). 

…two beams of light of different colours, what is different about them in the wave theory? And what is different about the photons…?

With photons, it's easy. What we see with our eyes as different colours are photons with different energy content in their field components.  And the energy content has been linked to frequencies with which the field components vibrate.
 With the wave it is less unambiguous and your question below somehow contains the answer why.

When we see two beams of light, one more intense than the other, what is different about them in the wave theory, and what is different about them in the photons that make them up?

More intense (for a monochrome light) is associated with a higher number of photons radiating into your meter or eyes. Unfortunately, the amplitude - which for water waves indicates the maximum height of the wave crest - was used in EM radiation as a synonym for the intensity of the beam. This has historical reasons from the experiments at the double slit.

When we say two sources of light are in-phase/ out of phase, how do we physically know this? And again, what is similar about two beams that are in phase in the wave model, and the particle model?

Again we have to refer to the double-slit experiments. Often it is mentioned that in front of the double slit a single opening will be the source of spatial coherent light:



    The first screen generates a point source,  so as to create a coherent wave . If it is a pin hole the geometry assures that all the photons come from the same original tiny source of light... Coherent means that the phases describing the mathematical form of the wave are not randomized.


The elaborated theory of double-slit fringes explains the swelling character of the intensity pattern behind the edges with the coherence of the incident light waves. Otherwise no pattern would form. To imagine this, one should try to draw or calculate monochrome waves phase-shifted with each phase shift for the right and left slots and it turns out that a blurred intensity distribution behind the edges will result.
The explanation with photons is more intuitive. If electrons in a source are accelerated in the same direction and with almost the same energy, they radiate polarized photons with nearly the same energy content, i.e. with the same direction of their E and M field components and of the same color respectively frequency.
It's harder with the phase. The best achievable monochromatic radiation is that of lasers. However, the phase of the outgoing photons depends on the laser cavity and the surface of the mirrors as well as the transparency of one of the mirrors. And never you will get a canceling out from two light sources like this is possible for sound waves. In sound waves the energy is dissipated into heat, for photons this is impossible. Photons are indivisible units form their emission to their absorption. A dissipation of two beams with 180° out of phase fields of the photons in the beam is not realized and hardly will be realized.
In short, we conclude from the slit experiments that light from two sources are in or out of phase. Could someone report about another experiment, to get results in and out of phase of light?
