If gravitons exists, then would there be anti-gravitons as well? 1)If not, why? 2) If yes, what are their expected properties?
I would have guessed the answer John Rennie gives, but I actually suddenly realize I do not understand why, so I sought to fix this lack of knowledge by some reading. You should wait for upvotes / my deletion of the answer to tell whether to trust it.
So, now, why is the postulated graviton its own antiparticle?
My understanding is that antiparticles are one particle states (irreducible representations in quantum state space of the Poincaré group see my answer here of what this means) that are mapped into one another by the CPT operator (or better written $C\circ P\circ T$) - time reversal followed by parity inversion followed by charge conjugation.
So now we need a sensible definition of when we call these two states "distinct" particle and anti-particle, or whether we say a particle is its own antiparticle as we do for the photon - charge conjugation is the identity map here and $P\circ T$ swaps left and right hand polarized one photon states.
The crucial difference between particles identified as their own antiparticles and the others then seems to be that self-anti particles can be created alone given some other quantum state with the requisite energy. momentum and angular momentum, whereas non-self-anti particles cannot be created alone with the requisite energy (because such a creation would violate the conservation of charge, or some other conserved quantum number) and must be created in $\psi$ and $C\circ P\circ T\,\psi$ pairs so as conserve electric charge or, I'm guessing more generally, generally conserved quantum numbers.
A graviton is postulated to be chargeless particle state. Moreover, it doesn't have any other quantum numbers (aside from spin) that would need to be conserved. Therefore, a quantum state with the requisite energy, momentum and angular momentum can evolve into a lone graviton without breaking known conservation laws. We would therefore call the graviton its own antiparticle, just as we do the photon.
This answer and the definitions assumed in it would seem to be supported by This Question and Answer Here and the links and discussions it cites, although there is not a great deal of discussion to be found there.