I think there are a couple of issues here. You bring up a good point that an alpha hydrogen that is also alpha to a halogen is more acidic, but this doesn't mean it will lead to a product.
1) Aldol addition where the ketone acts as an electrophile has a very small equilibrium constant. Only a tiny percentage of molecules will do this. A condensation product (alpha/beta unsaturation) might only form if water is rigorously removed.
2) Deprotonation of either alpha carbon is an equilibrium, reversible process.
3) Three membered rings form surprising quickly via Sn2, even though they are unstable, due to entropy factors.
4) The formation of the final product in Favorskii rearrangement is not reversible. Once it forms, it is trapped and will not go back.