Why must the graviton be spin-2 and how does it differ from other bosons?

[thierrykauf]

The electromagnetic field is an antisymmetric rank two tensor field therefore (not but) it is a vector field. Just count the number of components. Antisymmetric means its diagonal is zero. There remains one triangular block which has 3 + 2 + 1 = 6 independent components. These are the electric field and magnetic field vector.
Gravity is a symmetric rank two tensor, it's diagonal is not zero. It therefore has 6 (upper triangular block) + 4 (diagonal) = 10 independent components (the metric tensor). A spin n field has (n(
The solutions to Einstein's equations are NOT waves in general (no pun) since these are non-linear equations. But if you neglect nonlinear terms you can obtain wave-like solutions, those were detected recently. Nonlinear in this context means the graviton creates and is subject to gravitational attraction. That makes gravity similar to a non-Abelian gauge interaction such as QCD. In contrast EM is an Abelian gauge theory because the photon has no charge and therefore cannot interact with the electrons that produced it.

 

[vanhees71]

Of course, photons interact with electrons. That's how we detect them all the time.

Also your counting is misleading since both the photon and the graviton are massless fields and thus they have only two physical polarization states (except for scalar fields which have of course only 1).