All the conceptual difficulties in quantum gravity have a counterpart in the representation theory of the diffeomorphism group. This is not surprising, because the difficulties have to do with general covariance rather than with the specifics of the Einstein action, and the diffeomorphism group is the mathematics of general covariance.
The diff algebra in 1D is well understood - after quantization, we get the Virasoro algebra of string theory. The construction of Fock representations proceeds in three steps:
1. Start with a classical rep, i.e. a primary field = scalar density.
2. Introduce canonical momenta.
3. Normal order.
This recipe gives rise to a well-define rep of the Virasoro algebra.
However, it does not work in higher dimensions, because
1. In order to define normal ordering, we must single out a privileged energy or time direction. This gives us the usual problems with a foliation, which is at best ugly, and probably worse than that.
2. Normal ordering of a bilinear expression always gives rise to a central extension (central = commutes with everything). However, the Virasoro extension is not central, except in 1D.
3. It is ill defined. Normal ordering gives rise to an unrestricted sum over transverse degrees of freedom, which leads to an *infinite* central extension, i.e. nonsense. This does not happen in 1D, where there are no tranverse directions.
This problem is a manifestation of the usual infinities in QFT. In 1D, normal ordering is enough to remove them, but not so in higher dimension. I have tried to invent some kind of renormalization to remove these infinities, unsuccessfully. This is not so surprising, since GR is not renormalizable.
Instead, one must proceed along the following path:
1. Start with a classical rep, i.e. a tensor density.
2. Expand all fields in a Taylor series around a 1D curve, which I call "the observer's trajectory".
3. Truncate the Taylor series at order p.
4. Introduce canonical momenta, both for the Taylor coefficients and for the observer's trajectory.
5. Normal order wrt frequency.
This gives us a well-defined representation of an extension of the diffeomorphism algebra on a linear space. The reason why this prescription works is that in step 3 we have a classical realization on *finitely* many functions of a *single* variable (the parameter along the trajectory), which is precisely the situation where normal ordering works. The realization is non-linear, which leads to a non-central extension.
Let me write down the formula for the Taylor expansion, because it is the crucial new ingredient which resolves all problems. For every field f(x) (x is a point in spacetime), we write
f(x) = sum_m 1/m! f_m(t) (x-q(t))^m
where t is a single parameter, living on a circle, say. With multi-index notation, this formula makes sense in any number of dimensions. Instead of formulating physics in terms of the fields f(x), I use the Taylor coefficients f_m(t) and the observer's trajectory q(t). Classically, this is completely straightforward - the Euler-Lagrange equation for f(x) turns into a hierarchy of algebraic equations for f_m(t).
But we now have a privileged time variable t, and we can define the Fock vacuum by demanding that all negative-frequency components of f_m(t) and q(t) (and their canonical momenta) annihilate it. This gives us a notion of a privileged time variable without having to break covariance. Of course, we have added an extra dimension, and the RHS really defines a field f(x,t), so one must eventually impose some constraints on the f_m(t) to eliminate the t dependence. Nevertheless, f_m(t) still depends on t, even though f(x) does not, so the notion of lowest-energy representation still makes sense.
After this lengthy prelude, let us return to physics. The basic observation is that in any general-covariant theory, the Hilbert space carries a rep of the diffeomorphism group. Some clarifications:
1. For definiteness, consider the kinematical Hilbert space, on which the representation certainly is non-trivial. In the presence of anomalies, the rep must act non-trivially, and in addition unitary, on the physical Hilbert space as well.
2. In quantum theory, representations are always of lowest-weight type (the polymer reps of LQG apparently lack a lowest weight). This is true even for the kinematical Hilbert space, whose rep is not unitary, but it does have a lowest weight.
3. By diffeomorphisms I always mean space-time diffeomorphisms, which is the constraint algebra of GR in covariant formulations. Classically, the Dirac algebra is physically equivalent to the 4-diffeomorphism algebra, although they are mathematically distinct. However, apart from many other problems with the introduction of a foliation, I don't know how to construct reps of the Dirac algebra, so I have nothing to say about that.
In recent years I have tried to apply this math to the quantization of gravity. Unfortunately, none of the standard quantization schemes seems appropriate. The concepts from representation theory (action of a group on a Hilbert space) do not fit naturally into the path-integral formalism, whereas the Hamiltonian formalism breaks covariance which is precisely what the diffeomorphism group is all about. Moreover, there are big problems as soon as one deals with non-local objects, e.g. the Hamiltonian or the action which are integrals over space and spacetime. The Taylor expansion above is a manifestation of strong locality; the field f(x), which lives throughout spacetime, is replaced by data which live on the observer's trajectory only.
However, there is one formulation of dynamics which is both local and covariant, namely the Euler-Lagrange equation. Therefore, I have tried to invent a canonical quantization scheme based on this formulation of dynamics: manifestly covariant canonical quantization. It is in fact well known, from the days of Lagrange, that phase space is a covariant concept - it is the space of solutions to the Euler-Lagrange equations, modulo gauges (caveat about anomalies). My idea is to regard dynamics as a constraint on the space of virtual histories, and to quantize first and to constrain afterwards. In this way I can capitalize on the construction of diffeomorphism algebra modules, which describe virtual histories, without dynamics.
One should note that diffeomorphisms act in a well-defined way at every stage. This is because I primarily view the Hilbert space as a projective diffeomorphism algebra module.
By truncating the Taylor expansion at some finite order p, the normal-ordering infinites were removed. However, this is a regularization, and at the end one should take the limit p -> infinity; only infinite Taylor series can be identified with the original fields. It comes as no surprise that the infinities resurface in this limit. However, by considering more general representations starting from several fields, both bosonic and fermionic, one can cancel the infinite parts, but not the finite parts, of the anomalies. It turns out that this condition naturally singles out four spacetime dimensions. Although it would be a lie to claim that I predict 4D, this is still a promising hint.
There are of course also problems: I don't have an invariant inner product and thus no notion of unitarity, and one must probably develop perturbation theory within this formalism to actually compute something. There is also one rather serious flaw: classically, my construction gives the space of differential operators over phase space, rather than the desired space of functions over phase space equipped with the Poisson bracket. I think I know how to tackle these problems, but I haven't done so yet.
The most striking thing is the presence of the observer's trajectory. This leads to several conceptual observations:
1. Time is defined locally by the observer's trajectory, as the direction parallel to the vector q(t).
2. There is no global foliation, and no global notion of time. Everything is formulated in terms of the Taylor coefficients f_m(t), which live on the observer's trajectory.
3. The theory does hence not explicitly deal with objects away from the observer's worldline, except subtly in convergence issues, which I don't have anything to say about anyway.
4. The awkward notion of dynamical spacelikeness is also absent, since all data on the trajectory are causally related.
5. One can define a genuine local Hamiltonian, rather than a Hamiltonian constraint, as the generator which translates the fields relative to the observer, i.e. which acts on f_m(t) but not on q(t). There might be some problems here, though.
6. We can combine diff invariance with locality, in the sense of correlation functions depending on separation, but only in the presence of an anomaly. This is in fact well known in conformal field theory, which is covariant under analytic diffeomorphisms in one complex dimension. This is a important point, because the scaling operator does not depend on the metric, and anomalous dimensions are thus background independent.
It is getting late and this post is already too long. I am not sure how many of your points I have addressed, but this has to do for now.
And if it is not known yet whether you might dispose all along of second quantization or not (using the Barut self field approach), is fantastic news. 
