"I assume the math has something to do with the relationship between magnet field strength, turns of coil and rotational speed."
Yes that's correct. You should find formulae here somewhere. I have my own ways of calculating these things from first principles but others have given formulae that you will probably find easier.
Output power is more difficult. If you know open circuit volts, speed and winding resistance you can calculate the current you will get. It takes more experience to know if the thing will run at that current without frying.
You are aiming at high speed you will have a lot of mechanical issues to overcome but the size falls rapidly with speed for a given power.
Shape of coils doesn't matter much, basically use square coils with square magnets, round coils with round magnets and rectangular coils with rectangular magnets. You can compromise the shape on the inner radius to gain winding space without significant loss so you may find trapezoidal or oval coils used with rectangular magnets.
You can split the stator into 2 parts for different applications. Your big issue if this is variable speed will be regulating the output, these PMAs need electronics to regulate them effectively over a significant speed range, you have lost the absolute simplicity of field control with conventional alternators.
Flux
