Why do gravitons have spin 2?

The reason is that gravitation is described by a metric (symmetric 2-tensor field) modulo general covariance, which gives locally, in the tangent Minkowski space of any point, a spin 2 representation of the Poincare group modulo longitudinal directions, whch forces mass 0 and helicity 2.

Gravitational waves also have to be (classically) long range, whichagain requires (after quantization) massless particles.

Thus gravitons (although never observed) should be massless spin 2 particles.

Weinberg proved in

Quantization leads to a corresponding nonrenormalizable quantum theory. But it is perturbatively renormalizable if you allow (as in effective field theories) for an infinite number of counter terms suppressed by increasingly high powers of the Planck mass.

Today this no longer looks as strange as in the old days where the myth of non-quantizability of gravitation was coined.

Indeed, the predictive power is as large as that of a power series
whose coefficients are suppressed by a high power of the Planck mass,
where you only know the first few coefficients.

Arnold Neumaier (Arnold.Neumaier@univie.ac.at) A theoretical physics FAQ