Is there any way to shut down firefox if it is inactive? Some addtional tool or addon ? Or way?
Set your screen saver to 20 minutes or whatever you want, then instead of the normal screensaver just have it call pskill with the name firefox in the command line http://technet.microsoft.com/en-us/sysinternals/bb896683.aspx
Auto Shutdown: https://addons.mozilla.org/en-US/firefox/addon/5452/
There's a neat FF extension that's designed to enable a 'Kiosk' mode in firefox, with an idle-out feature, and more:
Not directly solving this problem as it seems like an overkill, but just a curious find...
Auto Close Idle is a program that monitors selected users' programs and closes them when certain amount of inactivity has been detected. Unfortunately, it is a commercial product and pretty expensive as it's intended to be used in domain environment, mainly to help reduce the amount of floating licenses needed.