9 V to 360 V via boost converter?
Can a charged inductor discharge current to a capacitor


In principle this is how a boost converter works but, to achieve 360 volts on the output, you need to rate your output switching transistor at something like 400 volts between collector and base / emitter. Unfortunately the 2N2222 is only rated at 30 volts so it likely will self destruct or limit the output to maybe 60 volts (and eventually fail).


  Can a charged inductor discharge current to a capacitor (connected via
  a diode so it doesn't oscillate)?


This circuit can easily become unstable and some care and attention on the layout is needed plus, ensuring the amplitude control loop is stable.