You would need to have an iTunes account.
Yes you can as long as your iphone is connected to the same itunes account you bought the apps on.
As long as your computer has itunes that is authorised on the same itunes account as your iphone, then it will automatically transfer all your iphone apps onto the computer
Some can be used on both devices, but some applications are for only the ipad and some are only for the iphone.
No apps are for iOS devices (iPhone, iPad, iPod touch) they don't work with other iPods.
Apple is too good for Android to take their apps directly. Their display would look weird on their devices anyway.
there's Derby Quest which is a horse-racing app for iOS devices.
iPhone apps work on iOS devices (iPhone, iPad or iPod Touch) they will not work on a Mac running Mac OS X 10.7 (Lion) or any other Mac OS X version
Yes, all apps work on both devices, the only exception are apps that use GPS since iPod Touch doesn't have GPS signal receiver (and older versions of both devices sometimes have trouble too).
You buy the iPhone app
apple create the apps on iPhone
Many apps will work on the iPhone, iPad and iPod Touch. Some are written just for the iPad while others only work on the iPhone and iPod Touch. Under the title in the App Store it will specify which devices the app will work on.