Someone must have stolen the catalytic converter  

Watch your voltage on the start, how low is it dropping during a battery starts vs a start stick vs a power cart.  If all 3 are appx same voltage drop then the starter is probably not the problem as it should make the same noise during the start.  When the static goes away note the NG speed and voltage, see which one is a constant for when the static goes away.  Check the main ground to airframe connection in the rear baggage, its behind the sheet metal but right where the EPU socket is, take grounds off, clean and retighten.  Be absolutely sure you know if the static is heard on all headset jack positions vs just the pilots.  Verify its on the ICS and not any radio circuits but having all sources selected off and rotary mic selector to a position that has no radio.  What model audio panel do you have, digital or analog? Are there 2 or 3 audio panel heads installed? Im guessing only 2 since its an AS350.