Spin conservation in pair production
Spin angular momentum is not conserved; only the sum of spin and orbital angular momentum is conserved. As a trivial example of this, consider a hydrogen atom decaying from $2p$ to $1s$ by emitting a photon. The photon carries one unit of angular momentum, but the spin of the electron doesn't change; instead orbital angular momentum is lost.
Furthermore, in many situations you can't even unambiguously define the two separately (how much of the proton's angular momentum is due to the angular momentum of its constituents?), so "conservation of spin" is not even meaningful. Conservation of total angular momentum is always meaningful, because it's the conserved quantity associated with rotational symmetry.
Based on the above argument, if the total spin in pair production is to be conserved, I would assume that the incoming photons must be in the spin-0 state, excluding the spin-2 state because the spin-state of the created electron-positron pair does not have a spin-2 representation. As far as I know, this spin state can have one spin-0 rep. and three spin-1 rep.
No, because the electron and positron can come out in the $p$-wave, carrying orbital angular momentum. This is called $p$-wave annihilation, and it's not an exotic phenomenon; for instance, it shows up in the partial wave expansion in undergraduate quantum mechanics.
Landau–Yang theorem, stating that a massive particle with spin 1 cannot decay into two photons. I suspect this selection rule follows from the requirement of the conservation of the total spin.
The Landau-Yang theorem doesn't state that spin is conserved. Essentially, it uses the fact that total angular momentum is conserved, along with the fact that in this simple situation, there is no orbital angular momentum: you can always go to the rest frame of the massive particle, and in that frame the photons always come out back to back.