Is my Rigowksi Coil turns calculation correct and how to calculate the sensitivity?
This is not a "real answer" yet.
It may become one.
But hopefully still useful.
I'm getting around to getting around to playing with Rogowski coils as part of a larger project. 

The internet is full of references as you know doubt know, but

Should be close to what is wanted DESIGN AND CALIBRATION OF ROGOWSKI COILS

This looks extremely good - despite the "relaying" focus:
IEEE PSRC Special report 2010 - Practical Aspects of Rogowski Coil
Applications to Relaying

More specific Design of a PCB Rogowski Coil based on the PEEC Method



Just for fun - Lightning measurement!!! with a Rogowski coil.