I wonder if the shadows from such a large area would have a climate cooling effect?

Ideally you'd put it in an orbit that wouldn't shadow earth, since if it goes in front of earth in part of the orbit then it'll go behind earth (and be in shadow) for part of the orbit. They mention they'd want it in a dawn-dusk sun-synchronous orbit, which would be in sunlight all the time and would never shadow the earth.

I think the principle is correct but probably it would need to be much bigger.