It is sustainable if the amount of biomass taken is less than the amount of biomass generated per area.

It is green in the sense that logging a forest net sequesters carbon from the atmosphere. Yes, even if you burn the wood. There's still carbon left in the ash and from discarded branches, and forests are most productive when regrowing.

You are right they are not healthy. The air pollution generated is nasty and even with a good catalyst stove cannot be reduced very much.