How can I duplicate Apple's tiny USB power adapter?
You're in luck: I just posted a detailed teardown and schematic of the iPhone charger. Internally it's a complex flyback quasi-resonant switching power supply controlled by a L6565.

Building your own switching power supply is probably more complexity and danger than you want to include in a project. I'd recommend going with the wall wart - there's a reason most products use one. If you really want a built-in power supply, I'd recommend getting a pre-built OEM one rather than building your own. And if you really want to build your own, I'd recommend a simple linear power supply instead of a switcher.