Does a fan whose blade geometry is optimised for generating airflow also have the optimal blade geometry for generating electricity if attached to a generator?

No, it does not. Blade geometry for generation is continually changed during operation to account for wind speed and direction. A fan assumes none of this, it’s more like a boat prop in the sense it assumes a static condition. Wind turbine blades operate more like the wings of an airplane.

No. A fan is designed to use mechanical force to push wind. The blades are designed to pull air in one way and push it out the other. But the mechanical force is the driving basis and the air offers little load. In fact most of the force of the motor is used to turn the blades with very little actually moving the air. In a fan/generator arrangement you are using wind as the motive force and the generator is the load. The generator is quite a load. You would need blades that are designed to take the force of the wind to turn it into mechanical force to turn the generator. As @Blackwater_Park said, that is more like the wings of an airplane.

No. Each are very different. Generating airflow versus generating electricity will have unique differing aspects. Here are a few but not all differences.
Number of blades.
Blade shape.
Turbine axis.
Tip shape.

I wonder what role the load (i.e. the hookup to the generator) would play. Would the blades need to pointed differently when under load?

Would the blades need to pointed differently when under load?

If you mean pointing into the wind, yes, the whole turbine always points into the wind like a weather vane.

If you mean blades twisting twist, to present a broader or thinner blade profile to the wind, yes, they do that, too.

“Pitch System
The pitch system adjusts the angle of the wind turbine’s blades with respect to the wind, controlling the rotor speed. By adjusting the angle of a turbine’s blades, the pitch system controls how much energy the blades can extract. The pitch system can also “feather” the blades, adjusting their angle so they do not produce force that would cause the rotor to spin. Feathering the blades slows the turbine’s rotor to prevent damage to the machine when wind speeds are too high for safe operation.”

Energy.gov – How a Wind Turbine Works