Constructing Gravitational waves with gravitons Suppose I want to construct a gravitational wave as a coherent sum of many gravitons. It's easy to think of what the frequency distribution of the gravitons should be, as all the LIGO discoveries more or less directly report this. But what would the number distribution of the gravitational wave field be? Would it be a coherent state, like a laser? A squeezed state? Something else?
 A: 
But what would the number distribution of the gravitational wave field be? Would it be a coherent state, like a laser? A squeezed state? Something else?

Hypothesizing  the existence of gravitons means that one has quantized gravity, and can describe it in quantum field theory as acting similar to all other gauge bosons in building up the classical analogue field. How photons build up the classical electromagnetic light in quantum field theory  is outlined here  and analogously the classical gravitational wave should be shown to evolve from gravitons, once gravity is definitively quantized. At present there are only effective field theories and thus guesses .
Classical is built up by the quantum underlying level by superposition of the wavefunctions of the quantum mechanical solution. The wavefunctions are complex functions and adding them up will display interference effects even though there are no interactions, just from the structure of the ensemble of photons ( or gravitons in this case). 
That this is true for photons can be seen in the interference effects of laser beams, even though there are no interactions of the photons. This MIT video (there is a bunch of videos)   is instructive. It demonstrates the effect of boundary and initial conditions, on how the photons superpose so as to generate a coherent beam.

Would it be a coherent state, like a laser? A squeezed state? Something else?

If you watched the laser video, you would understand the importance of the boundary conditions imposed on the beams. It is not possible to generate a single frequency, within the heisenberg uncertainty, gravitational field . The superposition principle will apply for all the spectrum of frequencies generated, for example by the annihilation of the two black holes into one at LIGO, and an incoherent beam will arrive with a frequency spectrum analogous to the incoherent beam of light hitting us from the sun. It will be mathematically more complicated, as graviton-graviton interactions are there , but the gravitational coupling is so very weak that it should not contribute in spoiling the above description. ( after all photon photon higher level interactions also exist, but the effect is too small to affect the light behavior in laser experiments or the sunlight).
Of course one must wait for a definitive quantization of gravity before accepting gravitons as more than hypothetical useful approximations.
