Classical solutions of Maxwell's equations describe light, electromagnetic radiation.
Electromagnetic waves can be imagined as a self-propagating transverse oscillating wave of electric and magnetic fields. This 3D animation shows a plane linearly polarized wave propagating from left to right. Note that the electric and magnetic fields in such a wave are in-phase with each other, reaching minima and maxima together
The photon belongs to the quantum mechanical frame, together with the electrons and neutrinos, see the table here of the elementary particles in the standard model of particle physics..
It can be derived mathematically that the classical electromagnetic radiation with the varying Electric and Magnetic fields emerges from a superposition of zillions of photons, where the frequency of the light gives the energy of the individual photon.
I.What does 'quantum' mean in this context? Quantum of what physical quantity?
Classical light has energy, the photon with its energy = to h*nu is a quantum of this energy. When light (EM radiation) is emitted, by atoms or charges,it is emitted as a number of photons, each photon a packet of energy building up the classical light.
II. What features do photons exhibit as a wave? (wavelength, speed, et cetera
The photon is a particle, i.e. it leaves a footprint when interacting with matter at an (x,y,z) like a classical particle would, and it is also a probability wave, i.e. many photons of the same energy passing through a double slit show an interference pattern, which is a wave property. Its speed is the velocity of light in vacuum, by construction of the standard model, and there has been no falsification of this "axiom".
on the left it is the individual footprints of photons, but the accumulations shows the classical light interference.
III. What features do photons exhibit as a particle? (mass, spin, et cetera)
Check the table. A photons mass is zero, it obeys special relativity rules as all elementary particles.
The momentum of the photon is given by the four vector that describes it , substituting in the special relativity equations.