Why can the graviton not be a spin 0 particle? On a similar note why can it not be a spin 4, spin 6 particle?
For force carriers the interacting field theory determines the spin. A scalar field yields spin 0; the Higgs is the only example; a vector field yields spin 1, the photon, W, and Z are examples; a tensor field yields spin 2.
Since gravitational field theory requires a tensor field for General Relativity, quantized gravity, in the weak-field, linearized limit, yields the spin 2 graviton.
Higher order spins can be constructed from complex particles, meaning systems with two or more constituents.
For more information see Would a spin-2 particle necessarily have to be a graviton?
Peter is right. The tensor nature requires it to be a spin 2 field, and the graviton is its presumed quanta.
But there have been and are theories of gravity that include a spin 0 field. Brans-Dicke theory was one (I think mostly or fully disporved), and some theories for dark energy are spin 0 - quintessence is one, it assumes the cosmological constant is not really constant but a slowly changing scalar field. Scalar fields, like the Higgs field, are spin 0, they just have a value at each point of spacetime. The field in that case can NOT be the same or some other entity associated with the Higgs field -- the Higgs particl is massive and is short range. Gravity is long range, and it's quanta is supposed to be zero mass or very light.
That dark energy field, in that theory, is called the dilation, it's what dilates the universe, accelerates it. The quintessence theory has not Been proven or disproven from measured or experimental data, but it is is believed that it is hard to match it To all the known experimental evidence for standard gravity which has been proven to sub cm ranges. It'd have to be such that the adjustment due to the scalar field does not conflict with known longer range measurements of planets orbits, the perihelion of mercury precession predicted by General Relativity, and many other. The standard cosmological model has that quantity, the cosmological constant, constant, not a field, and fits all the known data so far. More accurate measurements of the acceleration of the universe much earlier in its history will be done, but it's thought that it's going to be difficult to find enough of a change. That would prove it a field. Still, it is not known for sure either way. There are other theories of dark energy, none more or less likely as of now.
So, if dark energy is a scalar field there would be a Higgs like particle of spin 0, with 0 or very little mass, that would be the cause of dark energy. But it would not be a cousin of the Higgs.
Dark matter doesn't have the same kind of properties. It isn't everywhere, is strongly influenced by gravity from normal matter, and has been around since shortly after the Big Bang. It's believed to be composed of particles that are remnants from the Big Bang. It's gravity helped form galaxies and clusters/superclusters. It's believed that it is composed of some heavy particles or some set of them. It moves slowly, so they are not light particles. It's been mapped around and inside galaxies and clusters, so in some way it's easier to see (or feel it's gravity). Whatever it is is not Higgs like.
Spins greater than 2 are not possible in a theory that in inertial or local coordinates is Lorentz invariant.