I don't get it, if you can push the filament through the hotend by hand then it's not jammed, and it's hot enough to melt the filament. I would think the stepper would be able to do at least as good as it does by hand. And if the stepper was unable to push the filament it would just start to grind into the filament and slip, the motor would still be moving. So I wouldn't bet on a clogged hotend being the issue.
What does the LCD or pronterface report as the hotend temp? If the electronics is told the temp is too low it will prevent the motor from moving to prevent "cold extrusion". If you can extrude filament when pushing by hand the hotend is hot, apparently.... The LCD or pronterface will tell you the story there, if it reports hotend temp below the extrusion range (even though you can extrude by hand) then there is an issue with your temperature sensing circuit or an error in the config. In the config you could have the min/max temps for extrusion screwed up, or you could have the wrong thermistor model selected somehow, or there could be a wiring issue or solder issue with the hotend thermistor on the RAMPS board like I had (or a bad thermistor), or something else wrong with the RAMPS board.
As for cleaning, I always cleaned my J-head heater block/nozzle by heating it with a blow torch and melting/burning out any filament that is stuck in there (with the block removed from the plastic body of coarse). Then I would soak a q-tip in acetone and wash it out on the inside (after it was cooled of coarse), the nozzle always came perfectly clean after that. BUT I have never done this with my E3D head yet, I assume the same method would work...