Can Simon check that an application is still running on a different computer?

The Application service can only check for applications running on your local machine. But if you enable the Remote Login option in the Sharing system preferences on the other computer, you can use the SSH service to log in to the other computer and execute commands on it by modifying the SSH service to issue the "ps -A" command after logging in, then use filters to look for the application you want.

You can try it in the Terminal to see what such a session might be like, or just use Simon's Capture Session feature to generate the session.