I sought the answer to the question about amplitude of light waves first, but I was actually thinking about whether the wavelength is the only property of a single quanta of light. I suppose direction may be another, since it can be focused.
Now my question is about wavelength only...do photons actually have a wavelength? Or is that only a property that can be measured based on interactions with other particles (as in, how the photon can affect the state of an electron)?
You are asking about wavelength, but your question is a problem not very well-posed. Then, let's say so: from a single photon we cannot get a conclusion about it characteristics. Only if we know the beam to which the photon belongs, can we know the wavelength (or wavelengths see below).
Next, the wavelength is the result of preparation of the beam, it is not the result of the measuring procedure. In order to measure the wavelength, we perform different experiments, e.g. typically, interference, and for detection, the photons interact ultimately with all sort of materials, e.g. by the photoelectric effect. But, again, it's at preparation that we establish the wavelength.
Now, about the wavelength of a beam, it is rarely well defined, i.e. sharp. See a couple of examples:
1) thermal light, which has a mixture of wavelengths;
2) coherent light, which has a quite low spread in wavelength;
3) plane-waves, which have a well-definite wavelength, but are an idealization - not truly realizable in practice.
So, usually, the beam, and with it the photons in it, practically don't have a well-defined wavelength.
Yes, a photon is defined by its wavelength, which also directly relates to its energy.
I'm not sure what you mean by interacting with other particles, but a simple glass prism is able to separate the different wavelengths of white light into its constituent bands.
The wavelength of a photon of light is related to its energy by planck's constant.
