I suppose that you could describe Tomcat as "stateless". I never considered it in that light.
Tomcat isn't really the ideal container for non-web tasks. It does have the advantage of being a central nexus for different types of work, but that's about its only virtue for non-web functionality. You would be better off considering a simple non-web container such as a workflow engine, Enterprise Service Bus or OSGi container (these things tend to overlap). A place where I used to work used an Inversion of Control-based container descended from Apache's Avalon project. There were a whole raft of these things, although most of them have faded away now, and I never discovered what was supposed to fill the vacancy.
For scheduling, there's Quartz or the OS "cron" facility (Windows has a similar scheduler).
An IDE is no substitute for an Intelligent Developer.