As to the definition of fan vs. prop:
A fan is a part of the jet-engine compressor (among other things) - a propellor is not.
The turboshaft engine in a turboprop will perform absolutely normally with the prop removed (assuming one has a governor to prevent overspeed) - or with the prop held motionless, as with an ATR in "hotel" mode. A fanjet with the fan removed will be a degraded engine. And with the fan braked - won't run at all.
As to trying to make a prop work at faster speeds:
The usefulness of a propellor is that it moves more air,
more slowly, and is more fuel efficient in a low-to-moderate speed scenario. And requires a much smaller, simpler (read, cheaper) powerplant - turbine or piston.
As airspeed increases, a prop gets less efficient - regardless of whether it is ducted or not. In fact, ducting is most effective at low airspeeds (and high prop rpms):
Ducted fan - Wikipedia, the free encyclopedia (That link also lists other pros and cons of ducted fans/props).
Turbojets/turbofans get MORE efficient at higher speeds. So there is a cross-over point where it just makes more sense to simply use a turbofan, than keep trying to push the prop model faster.
If anything, the preferred approach, engineering-wise, to a "hybrid" engine is not to duct a prop, but to
unduct a fan.
Propfan - Wikipedia, the free encyclopedia
Although neither has really been successful in attracting customers, over the slow-prop/fast-jet model. Noise being one primary consideration.