(or any other kind of high frequency high voltage circuit.) why can you draw sparks off one side of the circuit when the other is not connected to anything and the circuit isn't grounded in any way (running off a battery for instance)? How does the current find its way back through the air?
Because of the electric field at the conductors. The field is strongest at the closest gap between them. If the field is strong enough the path becomes ionised allowing an arc to form.
Yes, but you get sparks if you touch just one wire of the inverter to a metal object (not even a grounded one nessecarily).
Do you mean grounded? The outputs are isolated from the low voltage side through the transformer so a grounded object would make no difference anyway. The sparks can be formed due to the capacitance of the metal object, and because the polarity is rapidly changing, there will be and arc between the metal object and HV wire as the plate charges/discharges.
