Do iPad apps run / work on the iPhone?
This depends on whether or not the app is a "universal" app for iPhone OS. Some applications are iPad-only and will not run on the iPhone. Others are designed to run on either device and will adjust display parameters and other functions according to where they are installed.
All iPhone apps will run on the iPad, whether they have been optimized for the iPad's larger screen or not. The iPad will run iPhone apps in a small window, or it can interpolate the app to expand and fit the iPad screen.