regulating voltage through Op Amp Non-Inverting amplifier
I think there is a pretty simple way. First, use an LM324 op-amp since it can run off a single supply. Power it with the 5 volts you are using everywhere else. Ground the inverting pin. Run your signal to the non-inverting pin and also to ground through a diode like a 1N4001.

The diode will clip the pulses from the coil at about 700mV and the op-amp is used as a comparator and will saturate to almost 5V whenever the + input is higher than the - input, producing nice 5V pulses.

If the output is on all the time you can use a resistor divider and the 5V supply with the - input to adjust the level at which the comparator flips positive. 

I am wondering about negative going pulses from the pickup. I would use two diodes in parallel (but facing opposite directions) across the pickup coil. This will limit the pulse to + or - 700mV and protect the rest of the circuit. Then perhaps AC couple to the + op-amp pin with a capacitor.