Re : It just doesn't seem to ring true that the climate system is so sensitive....
CO2 absorbs outbound infra-red radiation thus heating the atmosphere more than if it were not there. It's totally irrelevant whether this occurs at .003% or 1% - it depends on the gas. In the case of CO2 0.003% is significant.
However HEAT != temperature. Different materials have different specific heats and worse than that ice/water has a large energy involved in it's phase transition ( ice at 0C + lots of heat > water at 0C )