Driving highly inductive loads destroys mosfet driver

Holy Carp! You're trying to do 10's of nsec switching on a solderless breadboard? And you don't have a flyback diode on your transformer?

If you're going to do this stuff, you've got to learn to respect fast switching and inductive parasitics. Go to a ground plane and make all your switching paths as short as possible. Also, put a 100 uF cap (tantalum for choice) across your MCP1402 to give the flyback diode something to drive besides the long leads to the battery.

You see those regular bumps on your no-load waveform? They are ~40 MHz oscillations and they are not a good sign.

A combination of the IRF840's reverse transfer capacitance (120pF), the dv/dt of the drain voltage and the rather weak driver (MCP1402) is my best guess.

Starters, read the data sheet on the driver - it says on page 3 that the "latch-up protection withstand reverse current" is typically greater than 0.5 amps - that is a clue to why that device might be failing.

Next is Q = CV or, dq/dt = I = C dv/dt.

I'm thinking that the current through the 120pF with a big change in dv/dt on the drain is more than the driver can cope with. Just before the scope picture goes all bad I see something like 10V change in about 20ns therefore:

I = 120pF x 10V / 20ns - that is 60mA but that's just the voltage seen on the gate - it might by ten or a hundred times bigger on the drain and therefore the current might be 600mA to 6A forcing its way thru the reverse parasitic capacitor into the driver chip.

This is my suspicion anyway. I'd use a driver capable of ten amps or at least find one that can cope with a reverse current of ten amps.

Andy's on to something, I believe, with the drain-gate capacitance.

But also: Measure what this is doing to the 12V supply. That would be an alternative path for spikes through the gate driver. Currently you're showing a single 0.1uF capacitor as decoupling, and I suspect that isn't enough. You may need a broad spectrum of decoupling from 10nF up to 100 uF or more, and if that isn't enough, consider powering the gate driver and sensitive electronics from an L-C filter and their own local decoupling.