1) Does the GGG interaction of the EFT contribute to \alpha_s?
Remember than we are only looking at operators which will contribute at \mathcal{O}(1/\Lambda^2)!
EFT Concepts:
1) Why doesn't the operator H^\dagger \tau^a H \tilde{W}_{\mu\nu} B^{\mu\nu} interfere with SM in the cross section calculation for H -> 2 photons?
It has a different parity than the SM operators which! [Pointed out by Nick Kamp]
2) What's the smallest value of \Lambda_{\rm new\ physics} consistent with experiments?
In general, it's around 1 TeV. When looking for EFT operators which produce specific effects (e.g. in flavor physics), it can be much higher.