Electrical – PNP transistor with audio signal output on collector is used to drive opto-isolator. How does it work

A transistor will need about 0.65V to start conducting, so the iPhone's output is indeed too low. The lowest part solution to get a higher voltage will be the transformer. You'll need a winding ratio of 1:10 to get a decent output level.
You have to rectify the amplified signal. If you were just to average it using a capacitor you'd get 0V. You don't need full rectification, however. You can do without the negative part of the signal; there will always be enough positive signal to work with. Feed the signal via a diode to a capacitor. This will get you a peak detector: the capacitor will load to the positive peaks in the signal. Of course when the music stops the charge should also disappear. You can discharge the capacitor via the transistor's base resistor (missing in your drawing), and if you want to discharge faster have another resistor parallel to the capacitor.

For learning about op-amps and their problems, I'm tempted to suggest App Note 47; it's very long and focuses more on higher speed applications, but it has a lot of practical information on debugging this kind of system.

"How can you learn from experience if you can't identify the problem and just hope the circuit works the next time you attemt to build it?"

You have to localise the problem, even if that's just to one half of the circuit or another. You have to either build up from something known to work, or dismantle down to a definitely faulty core.

You need a model of the system, either a mental one or a simulation. Then you need to find out where the reality differs from the model. For audio stuff, this will involve poking around with an osciliscope a lot.