Ok. Here's some rough explanations for some of the things in your reply.

Wherever you have a quantum field, you have particles, at least for some set of observers in a relativistic quantum field theory. You were right that there are lots of subtleties in a complicated QFT like an extension of the Standard Model that includes CDM, but this generic property of relativistic QFTs is behind the term "particle dark matter".

CDM is "cold dark matter". Matter because it's a source of gravitation; cold because it moves very slowly compared to the speed of light (otherwise it would run away from galaxies rather than hanging around keeping them heavy); dark because it doesn't feel electromagnetism.

From the perspective of a relativist, CDM need not be particle dark matter. However, we can treat General Relativity as a field theory and can even quantize it (with some caveats) and the result gives us particles (gravitons in perturbative quantum gravity). New degrees of freedom that are "gravitational" are probably only "gravitational" in terms of the strength of interactions with other particles (i.e., very weak) but will still be representable as e.g. a gauge boson.

In a quantum field theory like the Standard Model, the fields' local contents are invariant under a set of transformations. When a configuration (like a proton) looks the same under various transformations, there is a symmetry at work preserving that invariant "look", much like when you rotate a circle around its centre in the Euclidean plane, it looks the same whatever angle you turn it.

The Standard Model has a number of parameters whose values are only known empirically, mostly by experiment, with little clue about why the values are the way they are. A common approach to reducing the number of such parameters involves expanding the symmetry group. Supersymmetry is one such approach, and creates a bunch of "superpartner" particles -- sparticles -- for each of the particles in the Standard Model. In minimalist supersymmetry theories, the lowest-mass sparticle tends to have the characteristics of Cold Dark Matter listed above. Other sparticles appear and can (via a new symmetry) explain apparent problems in the Standard Model.

The Ricci scalar is a value at each point in spacetime that describes the curviness of a shell of all points at the same distance from that point or equivalently the internal volume of such a shell. When the scalar increases at a point, the surface of the ball is locally flatter or equivalently the internal volume is smaller. If you move a ball through a region of spacetime where the Ricci scalar takes on various values, an observer at the centre of the shell will think the shell is more or less cramped as the value changes up and down, while a small observer standing on the outside of the shell will see the "ground" curve away (or the horizon move closer, equivalently) as the Ricci scalar decreases. If you seriously increase the Ricci scalar inside the Earth you would still measure a diameter of about 12 700 km, but from eye-level you could see to the other side of your continent. Likewise, if you seriously decrease the Ricci curvature below zero, you'd measure the same diameter of 12 700 km, but from eye-level your next door neighbour's house would be over the horizon.

The Ricci scalar appears on the gravity side of the Einstein Field Equations along with other quantities describing how lengths, durations and angles are calculated at each point.