The Earth's surface area is ~ 510,000,000 km^2. For simplicity, say exactly half of it is receiving sunlight. If the satellite perfectly shadows 16 km^2, it would reduce incoming solar energy reaching Earth by 16/205,000,000 = 0.00000078%, I think.

So not really. There are some proposals for space-based geoengineering, https://www.sciencedirect.com/science/article/pii/S146290112... has a summary. I can't say how plausible any of them are.

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.

Would be cool if so, but I wonder what the plan is regarding radiation pressure pushing things out of orbit.

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