Is it possible to programmatically close all running applications on the iPhone except my own from within my own application? If so, how?
It's not possible and there's also no need to.
There are very conservative limits on how much processing a background app can use and if the result of low memory warnings is that there still isn't enough memory, the OS will automatically start killing things. Don't try to be clever - it's just as willing to kill you and indeed will kill you first if nothing else is running.