I will explain it for an EFI engine with a manual transmission because it might not be as straight forward with a carbureted engine or automatic.

For ease of explaining this I will consider the engine as "on" whenever the ignition switch is set to the "on" state. This means that the ECU is on and is monitoring the engine state and controlling it's behavior. 

For the bicycle example with an EFI engine if you had your "ignition" to on while you started cycling then the ecu would start to inject fuel and cause spark so yes it would apply force and use fuel. If the ignition were off then the ecu would be off and you wouldn't get any fuel or spark so no you won't get any force if you apply the accelerator.

Going downhill not pressing the accelerator with the ignition on will use significantly less fuel (not sure if the ecu uses zero fuel though) and the engine will still be sparking for sure. If you were to turn the ignition to accessory while coasting downhill the ecu will shut off and you will stop injecting fuel and stop spark as well so the accelerator would no longer do anything.

If you have the clutch in and you stop the engine will stop. If you then release the brakes and roll down a hill then one of two things will happen.

1) if your ecu is on (ingition on) then the engine will start to use fuel and spark. (It will be really rough until it gets to ~700rpm)
2) if your ecu is off then the engine will rotate but no fuel or spark so therefore no acceleration.

Finally the difference between a running engine and not if they are both spinning :

A running engine is one which can continue to run from the energy it is producing without an external force. Modern engines require fuel air and spark to satisfy this requirement. eg. ECU on and rotating

An engine is off if when all external forces are removed then engine would stop. eg. the ecu is off so the engine isn't producing any of its own power and is only spinning because it is connected to wheels that are being accelerated by gravity.

Hope this clarifies it somewhat....

Does the a gasoline or diesel engine appy fuel at high speeds when not pressing the pedal and more
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