Does the graviton move at the speed of light in any reference frame?
Yes, provided that the speed (or more precisely, velocity) is defined in a general-relativistic covariant way.
Is there a Graviton?
In the standard model it says that there could be such thing as a graviton. But i would think that if a photon is moving at the cosmological constant (speed of light), then all elementary gauge bosons would be moving that same speed.
I would not insist on the term "graviton" as it contains some speculations about the quantization of gravity - which is by no means complete. Instead I would talk about a "gravitational wave", which always travels at the speed of light (provided that ... refer to Demystifier)
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