"On a wind turbine generator, when the wind speed is variable, what aspect of the power output from the generator changes, the voltage or the current?"
Ideally both will change but in your example only the current will change.
The charge controllers function is effectively to maintain the voltage constant. The battery does this until it is nearly fully charged and no controller is necessary. It is only with a full battery that the volts start to rise out of control to the detriment of the battery and the controller prevents this.
Your last bit of the question is too difficult to answer honestly but effectively your 120v generator will not be able to produce much more than nominal 48v. Its current will be limited to the 120v value for its own safety. I don't know where 13A comes from 640W at 120v is about 5.3A and it won't do much more for long is 640 is maximum rating.
What will happen in reality is very complex and we have no information to answer it. If it tries to produce much over 5.3A for long it will fry. Probably it will load the propeller down to a point where power is restricted and it may not even get to 5.3A
For normal operation the generator should make battery volts at cut in ( low wind) the actual generator voltage doesn't mean much as the battery will dictate what volts you get. At some point in higher winds the generator open circuit voltage will be in excess of your 120v and may be nearer 200 but you don't want this at cut in.
If your nominal 120v was at about twice cut in speed of your prop then it would be in the right order but in wind terminology we should consider it nearer a 50v generator.
These things are special, you don't use standard commercial generators intended for fixed speed working with an engine.