I don't understand how this element works. If it's just a "resistor", then
why don't we get electrocuted when we touch it and how can it arc with a
half inch gap opening the 220v circuit?


Mine was arcing to the oven; the oven is grounded, so you have a circuit.
Mine failed at the support that holds it up, so it just went to the support.
You are right; it can't arc over a half inch gap.

You don't get electrocuted (mainly because you don't touch it when it is on)
because the oven wiring is a much better ground than you, so essentially all
the current goes down the wiring and none down you.

And, why didn't a fuse blow? I know it's a three pronged grandfather plug
because that is what the firemen said when they yanked it out of the wall.

Why would it? The fuse will only blow if it the current exceeds the rating,
and there is no reason it should have. An arc fault breaker would have
popped, but you don't put those on your stove.
Does anyone know the answer to these questions?

These questions are easy; the one I can't answer is why it didn't stop when
you turned it off.

The fire extinquisher was ineffective because nothing was going to stop it
until you cut the power.