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It sounds like your thermistor is either loose or failing. Replace it and rerun the PID. Continuing to use it as is risks serious damage. 350c or more is no joke.
Z0 should be the nozzle just touching the bed.
Did you get a chance to look at the macros I linked above? There must be something missing or just a little off somewhere.
Are you probing with a heater bed or cold? Are you measuring nozzle height hot or cold?