A "particle" is a concept that only exists in classical mechanics. There it means that we are approximating the motion of an extended piece of matter (what we call "a body") by the trajectory of its center of mass, i.e. we are completely neglecting all internal degrees of freedom, the composition and we don't even care about rotations. Planets in the Kepler problem are "particles", photons are not.
One step up from "particles" reside the "rigid bodies", for which we deal with rotation and then, in continuum mechanics, we introduce deformations of elastic (and inelastic) solid bodies and "fluids", i.e. liquids and gases. Each time we include more degrees of freedom the mathematical description gets significantly more complicated and the phenomenological behavior of the corresponding matter gets richer.
Wave packets or "wavelets" are a somewhat vague concepts in classical field descriptions. They kind of seem to have a "center of mass", that kind of can look like they move like a particle if we squint on both eyes, but we almost never consider their proper behavior in media with dispersion because, at the end of the day, they are not as simple as particles and they don't behave in ways that make them as useful for applications in physics as the particle approximation.
For completeness sake, there is a mathematical procedure called a "wavelet transform" that works similar to a Fourier transform and that has very nice properties, but I am not sure if one can make it physically useful as it neglects the physical dynamics of the wavelets completely. They are merely used as kernels in a linear transformation, just like the periodic functions are used in Fourier transforms.
Finally, photons are not wave packets but quanta. Their main property is that they denote a finite amount of energy and, even more importantly, a certain amount of angular momentum that is being exchanged in the interaction between a quantum field and matter (which, strictly speaking, is also described by quantum fields). In this sense photons are "properties" or local measurements that describe dynamic processes. One should think of them similarly as one thinks about the conversion of kinetic and potential energy in classical mechanics. Instead of allowing arbitrary amounts of work being done by one system on another, quantum field theory limits it to small, but finite amounts... those are the "quanta". Quanta are therefor not "objects", but properties of objects or properties of the dynamics of objects.
From an experimental point of view a photon is the smallest measurement (or preparation) one can make on a quantized electromagnetic field because the measurement device is made of matter. One can therefor also think about photons as part of the state of the quantized electromagnetic field, but that, too, oversimplifies the mathematical framework that describes these field.
I. What does 'quantum' mean in this context? Quantum of what physical quantity?
Photons of frequency $\nu$ have energy $E = h\nu$. This means: if a photon of this frequency is absorbed or emitted, exactly this amount (quantum) of energy is transferred.
Photons of frequency $\nu$ have momentum $p = \frac{h\nu}{c}$. This means: if a photon of this frequency is absorbed or emitted, exactly this amount (quantum) of momentum is transferred. Photons move in the direction of their momentum vector.
Circularly polarized photons have angular momentum $\ell_z=\pm \hbar$. This means, if they are absorbed or emitted, exactly this amount (quantum) of angular momentum is transferred.
II. What features do photons exhibit as a wave? (wavelength, speed, et cetera)
As a wave they have frequency $\nu$ and wavelength $\lambda$, related by
$$ \nu\lambda = c,$$
where $c$ is the speed of light. The speed of light is the same in all inertial frames of reference. As a wave they also have an amplitude.
As waves they can interfere (e.g. double slit experiment), be reflected from mirrors, be transmitted through matter.
III. What features do photons exhibit as a particle? (mass, spin, et cetera)
The particle properties of photons are energy, momentum, and angular momentum. They have zero rest mass. Their spin (angular momentum) depends on their polarization, but it is integer, so they are bosons. As particles, they can, for example, hit objects and exert pressure. They can be emitted and absorbed transferring discrete portions of energy, momentum, and angular momentum.
Best Answer
Not really, no. That is a simplified picture but it's not the full thing.
A better statement is that a photon is a discrete excitation of a given mode of the electromagnetic field. If that (classical) field mode is monochromatic, then the frequency of the photon will be the frequency of the mode.
Note that it is also possible to have single-photon states that do not have a well-defined frequency, which are formed by taking a quantum-mechanical superposition of states with well-defined frequencies over a range of such frequencies.