Rain and climate change

I have been reading through as much stuff as I can find on the climate models, what I am not finding is anything that is predicting what will happen with precipitation over time.

With a higher temperature, there should on average be more evaporation of water and therefore more precipitation.

With a rising sea level, there will be a larger surface area to evaporate. A one centimeter rise in the oceans should lead to a significant increase in the water surface area.

With melting sea ice, there will be an increase in open water surface area and therefore more evaporation.

The increased evaporation should lead to increased cloud cover - what effect will this have? On the one hand it can trap heat on the other hand it reduces the solar radiation reaching the earth's surface.

Global warming should lead to a lot more rain over time, how will this rain distribute itself. It should lead to a higher degree of rainfall in a number of areas that currently are marginal or worse for rainfall. This could be a huge boon for the Sahel and even extend the Sahel northwards into the Sahara.

The higher rainfall should also be beneficial to a number of areas that have very marginal snowpacks at the moment. The higher temperatures and higher precipitation will in part manifest itself as a lot more snow in the boreal forest and tundra. Most of the tundra is a desert at the moment in part because it is too cold to allow for a lot of snow.