Goodness me, we'd better tell the aviation industry that it's incorrect to call a propellor a propellor when the throttle has been shut or the engine completely killed/died!
If you're actually connecting that propeller to a generator, then yes, it would be a turbine, however I'm not aware of anything that actually does that. There are often pop-out ram air turbines for that specific purpose, either to power weapons and other equipment on hardpoints, or as emergency backup power should your APU or batteries fail. In the case of a propeller, you would call them a retarder, and you typically feather your props to prevent that from happening.
Every prop can function as a turbine and every turbine as a prop.
Perhaps, but not efficiently. You can extract a whole lot more energy out of the flow with a turbine than you can put into the flow with a propeller. As such, a propeller is going to have mild camber allowing modest operation inverted as a turbine, but a turbine is going to be aggressively cambered and would immediately stall were you to attempt to use it as a propeller.