Where do conformal symmetry breaking and the gravitational constant come from in conformal theories of gravity? Conformal gravity theories are alternatives to GR which are conformally invariant. That is, if $g_{\mu\nu}$ is a metric solving the field equations of the theory, then so is $\Omega^2 g_{\mu\nu}$ for any nonvanishing function ("conformal factor") $\Omega$. Some people study the possibility that such a theory might actually be the correct theory of gravity at long distances, with GR being incorrect. One reason to think this might be true is that typically, such theories can explain galactic rotation curves without invoking dark matter at all.
The obvious first objection, to me, when thinking about such a proposal, is the fact that the universe does not seem to be conformally invariant. After all, in a conformally invariant theory, "distances" or "proper times" are really a kind of "gauge-dependent" quantity, which could always be shifted by using a metric which differs by a conformal factor. But in the actual universe, "proper times" of particles do seem to be real physical quantities. For instance, the probability that an unstable particle will decay is dependent on its lifetime and the amount of proper time it experiences. So this leads to the question:
Question 1: In a conformal theory of gravity, what breaks the conformal symmetry -- why does the spacetime metric seem to be a real physical quantity? Is it the Higgs which does this?
As a related question, at least in Weyl conformal gravity in 3+1 dimensions the coupling constant between gravity and the other forces is dimensionless -- so the Newtonian constant $G$ is not absolute.
Question 2: In a conformal theory of gravity, where does the "effective constant" $G$ come from?
I suppose the answer to either question might be "it depends on the theory", in which case I'd really like to see somewhere where this is addressed for one such theory -- say for Weyl conformal gravity or something similar.
 A: The point about conformal gravity is that vacuum field equations are conformally invariant. This means that when you have a solution of the vacuum equations and apply a conformal transformation, you generate the metric of a new vacuum space-time obeying the field equations. However, physical observers in this new space-time will observe, e.g., different tidal forces (geodesic deviation), since the Riemann curvature tensor is not invariant under conformal transforms and space-times are not generally Ricci flat in conformal gravity. In other words, there are quasi-local physical measurements that are able to distinguish between the space-times before and after the transform, and thus they are, physically, different space-times.
However, once massive fields are added to the action, the conformal symmetry of the equations is broken. In other words, if you take a non-vacuum space-time in conformal gravity with massive sources and apply a conformal transformation, you will get a space-time corresponding to different, possibly completely nonphysical matter sources.
Conventionally, one often talks about vacuum space-times which are vacua "almost everywhere", but the matter sources are still hidden in zero volumes as boundary conditions (e.g. black-holes"space-times, various "thin" disks and shells,...). The conformal transformation then turns the vacuum into vacuum, but almost always changes the meaning of the boundary conditions and thus of the matter source. (As an exercise, I recommend reading through Griffiths & Podolský Exact Space-times and finding out how many space-times can be conformally mapped onto the Einstein static universe.) In other words, the physical meaning of the space-time is generally changed by the conformal transformation.
You mentioned the Higgs' mechanism and it is, indeed, appropriate to mention it while talking about conformal gravity. The point is the Standard model has a conformal symmetry (the fields are massless) which is broken by spontaneous symmetry breaking (the fields become effectively massive). When you couple the Standard model to a conformal gravity, it has the same conformal symmetry, and this is broken by the SM fields becoming effectively massive. However, nothing special needs to happen in the gravity sector, it is really all about the mass generation in the source sector. To say at least, this is the scenario of how it should all work.
On the other hand, I should warn that conformal gravity has never been proven to be able to reproduce ordinary physics of the solar system, including not only the Newtonian $N$-body dynamics of the sun-planet-moon systems but also e.g. Mercury perihelion shifts, equivalence principle tests within the solar system and so on; post-Newtonian dynamics of binary pulsars, and others. In this sense trying too hard to assign physical interpretations to everything that appears in the theory may be futile, since it well may be that the theory has no physical meaning at all. In fact, the problem with the Newtonian limit of conformal gravity has been known since the end of the nineties, and nothing has really happened for decades now.

About $G$ "missing" in the action:
The Weyl-gravity term $\sim C^{\mu\nu\kappa\lambda}C_{\mu\nu\kappa\lambda}$ has the dimension of $1/L^4$ where $L$ is length, whereas the matter Lagrangian density is $\sim E/L^3 \sim m/L^3$ (suppressing $\sim c$ factors) where $m,E$ are mass, energy. As you have noted in the comments $[\hbar] = [E L]$ so one can write the coupling constant in the action as $(\hbar/\alpha) C^{\mu\nu\kappa\lambda}C_{\mu\nu\kappa\lambda}$ with $\alpha$ dimensionless and then it can be added to the matter part.
However, we still have to require that the theory has a Newtonian limit such that masses generate acceleration fields $\sim -\tilde{G}M/r^2$ in the weak-field Newtonian limit, such that $\tilde{G}$ is a factor that is generally quantitatively close to Newton's gravitational constant $G$ and $M$ is close to the dynamical mass of the object. Using only the constants appearing in the theory, it is obvious that in the emergence of the Newtonian limit there has to appear a new fundamental length scale $L_{\rm W}$ such that $\tilde{G} \propto L_{\rm W}^2/\hbar$. In other words, in the anticipation of the Newtonian limit, one could also write the coupling constant as $\hbar/\alpha = L_{\rm W}^2/G$. Picking between the two is just a matter of taste.
