No. Think of the home screen as just a place to conveniently place icons for apps you use on a regular basis... So you don't have to go hunting for them in the App Drawer which contains icons for all of them.
As to if a program is running or not, that is a function of the OS and if it decides to start it, or you start it. It might pre-load it into memory in anticipation of you running it.
To see what is running on your phone, you can either download an App Manager, or use the built-in one: Menu | Settings | Applications | Manage Applications. With that you can see what has been downloaded, what is running, if it has been moved to the SD card, etc.
Hope that helps.