Does a quadrupole transition mean emission of one photon with spin 2? If it's true and spin-2 photons do exist, could you please point to some literature that discusses spin-2 photons?
If not, then how exactly does a selection rule for quadrupole transition make sense in terms of interaction with spin-1 photons?
 A: No, photons are always spin-one particles. The state of a photon may be described as a linear superposition of states with well-defined momenta $p$ and polarization vectors $\vec e$ and each of these basis vectors corresponds to a spin-one particle because the polarization is given by a transverse vector.
The selection rules – see this summary table – primarily boil down to the angular momentum conservation law. The conservation law says that the angular momentum of the atom before the transition is equal to the angular momentum of the products after the transition (the atom in the final state plus the photon). So for quadrupole transitions, as the table recalls, we may have $\Delta J = 0,\pm 1,\pm 2$ – depending on the relative angle between the angular momenta.
There isn't any need for the photon itself to have $S=2$ just because the operator responsible for the transition is a spin-two (quadrupole) operator. Such an agreement between the angular momentum of the operator causing the transition on one side and spin of the photon on the other hand would only have to hold if the initial and final states were the same – or at least had the same spin.
But because in the transition, the initial and final state of the atom are different and have different values of the spin in general, the spin-two character of the transition operator (quadrupole) means that the photon plus the difference between the final and initial spin of the atom corresponds to spin-two: this conservation law doesn't constrain the spin of the photon itself.
To summarize, I believe that in your argument which was "nearly" right, you forgot to add the difference between the final and initial angular momentum of the atom which is nonzero.
A: Photons always have spin equal to 1, just like electrons always have spin-1/2. The spin is a fundamental property of the particles.
To make sense of the selection rule for quadrupole transitions ($ \Delta J = 0,\pm1,\pm2 $) you need to remember that the photon can also have orbital angular momentum in addition to its spin. For example, in an electric quadrupole transition the photon can have orbital angular momentum $ L = 1 $ which adds to the spin angular momentum $ S = 1 $ to give total angular momentum $ J = 1 \;\mathrm{or}\;  2$.
A: I assume that in the quadrupole transition there is a 2 photon interaction, since photons only have spin 1. Hence why it is a less probable transition.
