Is there any quantum-gravity theory that has flat space-time and gravitons? Many quantum-gravity theories are strongly interacting. It is not clear
if they produce the gravity as we know it at low energies. So I wonder, is there
any quantum-gravity theory that
a) is a well defined quantum theory at non-perturbative level (ie can be put in a computer to simulate it, at least in principle). 
b) produces nearly flat space-time.
c) contains gravitons as the ONLY low energy excitations. (ie the helicity $\pm 2$ modes are the ONLY low energy excitations.)
We may replace c) by
c') contains gravitons and photons as the ONLY low energy excitations. (ie the helicity $\pm 2$ and $\pm 1$ modes are the ONLY low energy excitations.
This is the situation in our universe.)
 A: A well-defined quantum theory is clearly presented by Rovelli in the 2011 Zakopane lectures: http://arxiv.org/abs/1102.3660
It definitely satisfies your criterion A, easily seen to heuristically give B, and I do not know personally what is the status of C, but I know that a graviton propagator is definable and computable, which might be sufficient.
Personally, I believe there is a lot of underlying commonality with your own work (which I follow with a dilettantish interest). In particular, Rovelli has also introduced fermion and gauge theory coupling by means of lattice field theory living on a quantum graph, which to my mind resembles string-nets: http://arxiv.org/abs/1012.4719
There are also a nice set of recorded lectures at the Perimeter (perhaps you've already seen them in person, however), which contains a lot of colloquial talking which helps to fill in between the lines, and which I think expresses Rovelli's personal view of the state of the research much better than his written work: http://pirsa.org/C12012
A: I think the answer to your question is "No."
String theory seems closest to me, since one can formulate it in terms of matrix models, which resemble models we do know how to simulate.  One can also make computations in this fashion which match with supergravity computations (e.g., Robert Helling's early work).  I'm not sure, however, that it's known precisely how one would express a Standard Model computation in terms of matrix models variables.
A: Asymptotically Safe Gravity (if it exists) should have that pretty much by definition: In the low energy limit it reproduces the Einstein-Hilbert action which has flat space as a solution around which you can consider small perturbations, in the high energy limit you run to the fixpoint. 
