Concave RD hanger
 

I removed the hanger from my bike and put it against a bunch of flat things, such as my cell phone screen, some well-made allen keys, and my desk.

It appears a bit concave in the center. The bolt holes and the other end where the RD threads line up. It's just that the middle of the RD hanger has a very mild and rather smooth curve in it.

Normal? By design? Or what?

There's really no way to tell.
As long as the derailer mounting surface is parallell with the hanger/frame interface surfaces, all is well.
The shape inbetween doesn't matter.

Remount the derailleur hanger and then put a Park DAG-2 or similar tool on the bike and see whether or not it is out of alignment. If it is within the +/- 4mm tolerance specified by the instructions that come with the tool and your shifting is otherwise ok then remount the derailleur and just ride.

-j