What exactly is meant by the wavelength of a photon?

A photon is a measurement on a quantum field. It's a "one time deal", if you like. Each photon has an energy and a helicity (sometimes confused with "spin"), but that's not enough to produce a "wavelength", which is a property of a classical electromagnetic wave. We only recover the wave by measuring many photons, which then approximate the classical wave shape. In order to have a sensible definition of wavelength, these photons all have to have a similar energy, so that the coherence length of the resulting wave is long (enough). Strictly speaking one would not assign that wavelength to the single photon since the single photon measurement can't tell us that the wave is sufficiently coherent.

Concerning massless particles, don't forget that the spacetime of their lightlike worldline is empty (= zero). That means that the point of emission and of absorption are adjacent in spacetime, even if the space interval between them measures billions of lightyears. By consequence, there is no problem for the transmission of particle characteristics for massless particles.

The wave of a photon is propagating through space with velocity c, and the lenght of a wave is what we can measure in space (with a meterstick), even if the spacetime interval is zero.

This rule does not apply to photons moving at speed v < c through matter. The particle characteristics are transmitted, but the spacetime interval of the worldline of their timelike movement with speed v < c is not empty. This is one of the phenomena of quantum nonlocality, and we can only describe and calculate it, but we haven't got an explanation yet.

It had been known for a long time that light showed interference effects, just as in other waves such as sound and water waves. So in the double slit experiment with monochromatic light you get the light and dark bands on the screen, from which you can work out a wavelength for the light. It was thus assumed that light being a wave there must be a vibrating medium to transmit it, which they called the ether. The big difference between light and other known waves was that for light there was no alternating physical phenomenon such as the height of water or pressure of air which for classical waves could be directly measured. The ether theory was knocked on the head by Planck and Einstein when light became a particle, and Max Born gave the only possible feasible explanation of the interference effect, that the wave property (the square of the modulus of the complex number obtained by adding up the various possible paths) determines the probability of the photon landing at that point on the screen. Its as if nature had been fooling us into believing that light must be a classical wave, when all along the meaning of 'wavelength' and 'interference' are quite different to a classical wave. In Feynman's book 'QED' he talks about 'arrows' which are complex numbers represented in the complex plane which are rotating according to the frequency of the photon, thus describing a spiral as the photon moves along. The wavelength is the distance for which the arrow goes once around. Its a mathematical device (the complex plane does not exist as a real object) that however gives us the results of real experiments.