My iphone 4 does have itunes on it. I know to tap pause to stop the music, but it seems funny that even when you press the off button on the iphone, the music continues.
Minor nit, but iTunes is the application on your computer -- what you're referring to is the iPod application on your iPhone.
Also, when you press the "off" button that is actually the Sleep/ Wake button. Pushing this button does not turn the phone completely off, it just shuts the display off (any applications will continue to run in the background). To turn the phone off you have to press and hold this button until you see a red arrow, then slide the arrow over to shut the phone down -- that is OFF, what you're referring to is Sleep.
No it does not. Your iPhone includes a iPod, which is not iTunes as in the iTunes app on your computer.
Even the original iPod was not called mobile/portable iTunes. It is called an iPod, and the iPhone includes an iPod.
Pressing the off button by itself does not power the iPhone off. You must press and hold the on/off button until the slide to power off is provided followed by using the slider to power the iPhone off.