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It is no secret that white reflects heat, and that is true for your roof as well. In fact, many hot climates around the world have been utilizing this knowledge for centuries.
βThe darker the surface, the more the heating. Fresh asphalt reflects only 4 percent of sunlight compared to as much as 25 percent for natural grassland and up to 90 percent for a white surface such as fresh snow.β Yale
Other places have also started catching on. For example, even in 2012 in the USA, volunteers started painting tar roofs white in New York. However, this is not something trending everywhere. Some places are accustomed to dark colored roofs, so even in hot climates, they are slow to change.
This change will of course be easier in places where flat roofs are more common and ready to paint, but slanted roofs have options too.
For example, metal roofing is more environmentally friendly than shingles(100% recyclable), is more durable, and can last longer. What is more, the reflective properties of the metal can also help cool your homes. Some are now even made to look like tiles.
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There are other (more expensive) options as well, such as painted terracotta.
However, if these are not your style, solar panels on your roof may also reflect heat, so you could generate some power while you stay cooler.
βSolar panels βcool daytime temperatures in a way similar to increasing albedo via white roofs,β according to a study by scientists at the University of New South Wales. βYale
If you live somewhere cooler, I do not suggest a white roof, since the white can raise winter heating bills.
IIRC in India they used a kind of white paint that additionally absorbs co2.
If you mean slaked lime, that needs to be first burned, which releases CO2, so that ends up at best CO2 neutral.
I don't know. It was something experimental.