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It at least doubles the thrust per swept area area, probably more, due to the counter-rotation and that the second stage rotor experiences a higher airspeed.

This isn't a multicopter. It's a coaxial helicopter. The design parameter most similar to the distance between two rotors on a multicopter is the rotor diameter.

It's obviously going to be less maneuverable though relative to the thrust, due to the cyclic being on only one rotor.

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Far less than double. Putting one rotor directly in the turbulent wash of another is nowhere near as efficient as two well-separated rotors, for a number of reasons. https://en.wikipedia.org/wiki/Contra-rotating_propellers gives an estimate of "between 6% and 16% more efficient than normal propellers".

Yes, I don't know why I wrote it, since it's obviously wrong. I assumed mentally that they'd put as much power into the second propeller as into the first and that it'd go into the airflow reasonably effectively, but you don't double the thrust by doing that, you presumably double the power in the airflow, so that its velocity velocity to sqrt(2) times what it was, so you get a momentum increase by sqrt(2) and since force is the derivative of momentum the thrust is increased by sqrt(2), and then you a thrust increase by sqrt(2), and maybe you also get those 6-16% you mention.

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