Will a faster rate of change of flux improve electricity generation?
In theory, yes. 
Practically, your idea makes no sense for the following reasons: 

Your basic idea is right, faster variation of flux means bigger induced current, quite simply. Of course in reality this is way more complicated and there are a huge number of restraints. Also, the mechanical power input must always exceed the electrical power output. Not like you can magically create twice as much electrical power with the same input. 

The idea is, having magnets pass changes in the field more often. Practically, this is done by giving the generator more poles. This means the same: more changes per revolution. 

Rotating both magnet and coil at the same time is a bad idea for several reasons. 
In big installments, it is a challenge to rotate even one. Having both rotate coaxially adds even more of a challenge, so much so it is basically idiotic to try. How do you apply mechanical power to both, etc? In power stations, the generator is also set up in a way to have the coils standing fixed, avoiding the need to grab huge currents from a rotating part. 

Also, power generation is generally synchronous, 50 or 60 Hertz, so it is inadvisable to change the amount of fluctuations per second. The setup for the generators is very much fixed.