If I plug an iPhone that's running an old version of iOS into a computer with iTunes, will iOS on the iPhone automatically update when the phone is plugged in?
No, your iPhone won't automatically update its software when connected to iTunes. You get an alert that your iPhone can be updated to newer version of iOS, but you can just close out of the window that the alert is in. Your iPhone may, however, automatically sync with iTunes when connected to your computer, but this can be disabled by unchecking a box in iTunes.
You'll get a notification to download and install it. You can opt out of this though.