A rocket engine doesn't require combustion, it just requires expelling a gas to produce thrust, and that the engine not be air breathing. Most rocket engines use combustion to produce the high speed gases, but there are plenty that don't (water rockets, steam rockets, plasma rockets, and... cold gas thrusters as demonstrated by Tesla)

Motors don't need combustion, they just need to provide power. See electric motors or compressed air engines.

I doubt it is actual air. When compressing gas to 10,000+ psi and then running it through a rocket motor, clean/dry nitrogen is the way to go. Normal air has moisture, which can form ice, and oxygen which gets extremely reactive when heated.

And many a rocket scientist would be insulted by the comparison of a nozzled rocket motor to the simple pipe of an air rifle.