Yes, you can absolutely copy and paste on an iPhone. Usually, you just have to tap twice and options including Cut, Copy and Paste will display. In most apps, there will be handles around a word or words which you can adjust to tell it what you want to copy; then when you are in another email, note or other app, just tap twice again, and you should see the Paste option. In some apps, the initial double tap will display different options. For example, in Notes, when you double tap, it displays options to Copy, or Select All, along with handles to adjust the text selection.
In text messages, double tapping only gives me the option to Copy; then when I tap in a new text message, double-tapping lets me Paste.
By the way, I'm using an iPhone 4S.
Hope that helps.