Windows shared server quick start app pool
Currently from my chat with tech support, there is no way other than contacting someone to start the app pool once it stops. I'm not asking for the functionality to stop/restart the app pool as I can understand how that would affect other websites on a shared host. I feel like it would be super useful to have a button to do a start on the app pool if it has indeed stopped to avoid downtime while requesting the app pool be started through support.

At the moment you can start/stop/recycle application pool in Tools&Settings>IIS Applications Pool. In upcoming Plesk 12/5 you will be automatically offered start app pool when you open this page in case if it is stopped for some reasons.
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IG