Afaik weather will become more extreme, so some regions will get very wet (flooding damage) (as we see currently with Portugal), and some very dry (drought).

From a physical viewpoint, climate change will heat up the oceans, so more energy will be available to lift up water into the atmosphere/clouds?

From what I've read on this, hot-dry places will get hotter and drier, with intense desertification. Places that are typically wetter will have more frequent bouts of flooding and unusual rainfall patterns, like places that traditionally had "balmy' weather might experience sudden 40day+ continuous heavy rainfall.

Also, even though on a whole the world average temperature will increase, cold weather will also tend towards the extremes with longer, colder periods.

Here you can see the effect of increasing global temperature on precipitation:

https://interactive-atlas.ipcc.ch/

Yes and no, sometimes one after the other. Climate change is more about the climate destabilizing and becoming more erratic as opposed to moving in any particular direction.

For rain to be useful it needs to be consistent and predictable.

If it only rains on 1 day a year, and on that day you receive 1 years worth of rain, it just washes all the soil away.

More extreme weather

Building rainwater-storage-facilities for such sudden changes in rain patterns is hard as well.