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)!
2) In a term of the EFT with |H|^2, how can one replace one of the Higgs field with its vev and not both?
Remember that what we are actually doing is writing the lower component of the Higgs field as h+v, where h is a dynamical, real Higgs field and v is the vev. So, when we expand out a term quadratic in the Higgs field, we get terms with h^2, v^2 and hv. Thus, it is fair to think about replacing only one of the fields with the vev.
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.