Engine\belt power has absolutely nothing to do with angular momentum. What matters is the inertia and RPM of the belt. What is the formula for angular momentum?
This is not correct. I don't know where you got four pi times the integral of the resistance across the surface from.
I don't want to waste your money on a wild goose chase, so I would look over these slides instead for now:
No, an engine powers a propeller. The engine can be reciprocating or a turbine. There is no such thing as a 'propeller engine'. (Or it can be an electric motor)