A modern jet engine is just a shrouded propeller that happens to be powered by a jet turbine. An electric ducted fan would be the battery-powered equivalent.
No, it's not. A propellor creates thrust by slicing the air and creating lift just like a wing, but in a forward direction. A jet engine's fan blades create no thrust in themselves. They provide the compression needed for combustion to occur within the engine, and thrust is created by the expelling of hot exhaust gasses.
This is done to trade the high speed of the jet exhaust for a larger amount of air moving more slowly. Making the speed of the jet plume closer to the true airspeed of the aircraft makes it more efficient.
Looking at design of turbines, a lot of air is needed above the amount for combustion to help keep the post combustion chamber turbine from melting (or just weakening to the point of failure). Some of this will be the unburnt (nitrogen, etc) components of the ingested air but a lot of design goes into moving air around components to move heat away from them.
The less air needs to be compressed for the engine then the more efficiently it will run. This is often achieved by designing one that can run hotter. This is also one of the other ideas behind having the bypass on a turbofan, reducing the amount of air that is compressed.