How can I run another app when pressing the calculator key on my keyboard?

Edit this registry key:

HKLM/software/microsoft/windows/currentversion/explorer/appkey/18/ShellExecute

If it does not exist, create a String Value with name ShellExecute and as value the path to your program.

Did you try Logitech's SetPoint software? It may not let you change what program the key opens, but it's worth a try. I don't have a Logitech keyboard, so I can't check.

You could also try replacing the Calculator executable (%windir%\system32\calc.exe) with the other calculator, but that may be risky.