I want to supply my own stand alone application manager, but I want my application managable by enterprise managers too. Can I solve both of these problems with JMX technology?

Yes, and you even have choices on how to do this. You can instrument manageability in your application with JMX technology. Your application manager can use its own JMX technology adapter to manage the application - the same way the enterprise managers do! Alternatively, your application manager can expose the managed resources and management interfaces to JMX on behalf of your entire application. Obviously, with JMX technology, you can have the best of both worlds.