A few questions regarding the functionality of this negative peak audio detector
Why does the output of the op-amp (purple trace) [initially] stabilise at -0.6V?
The feedback loop (R at the top) compensates for the diode drop.
Why does the output finally go to zero?
This will depend on the diode model used. A good model will conduct some current at very low voltages and eventually have a drop of zero.
Are the caps in the wrong direction?
Non-polarized caps would normally be used here (better tolerances). I realize that you didn't draw this, but when drawing your own schematics, please don't use that symbol for polarized caps, it is not widely used anymore and not everyone will immediately recognize it. Add a plus sign for polarized caps.
What is the purpose of D5?
D5 keeps the opamp out of saturation (when the input is + for a negative peak detector). It takes time to recover from saturation and part of the next valid input signal can be missed without it.