Electronic – How to connect a laser diode

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.

A laser diode is usually a three terminal device: a common point, a supply pin for power to the laser diode itself, and a photodiode output for feedback. The device you have looks like it has either a built-in controller or is running in straight open-loop (uncontrolled) mode.

If you keep the current within the Safe Operating Area (SOA) for the laser diode, you won't hurt anything, but you end up with an uncontrolled laser. It's not as scary as it sounds, but you probably aren't lasing or, if you are, you have no idea what your actual laser power output level is, which can be dangerous to eyeballs.

Laser diodes are designed to be operated in closed-loop mode where the power supply senses the output level of the laser through that photodiode and adjusts its output current on the fly in order to keep the device laser diode operating in the region where it is actually a laser and also to keep the power output level where it is designed to be. The link you provided isn't a laser power supply in that sense; it's a simple open-loop power supply which is probably better at maintaining a specific voltage/current level, but not actually maintaining a specific laser output power level.