HDR is more about optimizing screen setting than adding new elements. But what about games? 

You can't just magically add it without something processing it, so in short yes it requires extra processing power. Whether that extra is negligible or not is a case by case situation.

I very much doubt it will take much since it mostly allows a greater variety of pixels (colour and brightness) to be shown. It doesn't take much power to render those pixels. I don't believe I have seen regular PS4 games run worse because hdr was enabled. So I don't think it does since it simply allows more sorts of pixels being able to be rendered. The game itself isn't rendering them differently it might take a little on bandwith. Developers (sucker punch, naughty dog and guerilla games) said implying hdr is easy and non demanding.

Isnt it more about how much space things take up? not sure if it actually requires that much (if any) extra processing power.

Probably minimal, if at all. HDR is just allowing a wider color gamut and greater contrast (think different video profile). That does not require more GPU to push more pixels or render higher resolution textures. I am sure digital foundry will do an analysis on it at some point.

The "extra processing power" needed will be as high as running a PC game in 32-bit color mode instead of 16-bit color mode... there will be almost no difference.

No, or at least a completely negligible amount of power.

Some more is needed, but it just affects some stages of the graphics pipeline, and obviously it doesn't affect almost at all the CPU. Obviously the resource that it needs more is memory.