Now I need to monitor the voltage of the discharge delay curve of the coil shortly after 5us as shown in this photo.
No chance to monitor the discharge curve directly, a single measurement with the ADC takes ~110µs. This can be improved to around 10µs, but you would need another order of magnitude faster for that curve.
If instead you create some kind of LC oscillator whose frequency changes based on the metal, you'd be able to measure that frequency with an Arduino (it can handle up to clk/2, so 8 MHz for a 16 MHz Arduino - a 10 µs period is 100 kHz). Such oscillators are easy to build with a series of NOT gates, and remarkably sensitive to changes in electromagnetic fields. I think that'd be a good starting point for a metal detector.
For the circuit in the OP, and that question: it's trivial to have an Arduino produce a tone output based on the analog input:
produces a tone of 500-1523 Hz on pin 5 depending on the analog reading on A0.