I know I am late to input anything here, so sorry up front.
Your transistors seem to die because they are loosing power to the base, the resonance still stresses the collector and emitter.
So get a second supply for driving the base - also (for not seeing the setup) consider switching to mosfets and put a 12 volt ZD on the base ( edit base should say Gate) to ground.
I skipped most of this tread, as time is limited, I can see what you started saying about charging and discharging the cell itself.
Did you connect a wire to the outside of the cell (with foil around the outside cell?), I read the first few pages looking for a photo, then skipped to the last few pages, will get to the middle soon.
warj1990