Scenario 1:
I need to increase RPM before transmitting motion to a generator. Is a gearbox more efficient and cost effective, or should I use a hydraulic method: transfer the motion to a radial piston pump, which outputs high pressure fluid, then use a hydraulic turbine connected to the same generator.

Scenario 2:
Just in general, what are the advantages of using mechanical systems over hydraulic, assuming the overall functionality is more complex than just a pump for example.

 

I believe a simple gearbox is typically far more efficient than transfer via a hydraulic system. With a simple spur gearbox, you're probably looking at somewhere around 99%+ efficiency, whereas a hydraulic system is likely in the 80%-90% range.

Hydraulics are really great if you're generating large forces in a small system, covering long distances, or serving many devices from a single source.