1. Could you explain relations between independent components of the metric tensor field, its helicity, polarization and spin?
Let me specify the question. The metric tensor in 4D has 10 independent components. From dynamical analysis of the Einstein-Hilbert action we know that gravity is a theory with 2 degrees of freedom. Quantum field theory tells us that it describes a spin 2 particle, thus has five helicity modes ±2, ±1, 0. It also tells us that the graviton is massless (let us neglect massive graviton models if possible), thus only two of these modes are physical, ±2 ones. I could be wrong in this statement, but that issue looks analogously to non physical polarizations (longitudinal and time/scalar) of a photon in contrary to the massive Proca field. Finally, some theories extend dimensionality of the phase space of the gravitational field to more than 10*2 dimensions. For instance, considering Ashtekar variables, we introduce additional internal SU(2) symmetry in the spatial sector of the metric tensor, obtaining (3*3+4)*2 instead of just mentioned (6+4)*2 dimensions. I would not try even to interpret these extra non physical degrees of freedom and the corresponding gauge, but maybe someone could do it.
2. When explaining graviton quantum numbers could you refer to the specification of the problem above, giving analogous examples for photon and weak bosons states (even considering a massive gravity for the latters if necessary)?