When you duplicate a published app, a copy of this app is created in your "My Works" folder and this is now a completely different app and it won't have direct connection to your preceding tasks.
What you can do is to develop on this new copy in "My Works" folder and when your development has finished, you can publish this app onto your existing published app while ticking "Replace existing app" option.
Your previous task settings will still be valid.
no, the task isn't copied too.
Each application as a unique ID (which you can see in the application list). On this ID the tasks and dependencies is on. If you copy your application, it gets a new ID.
If you copy your application and republish them within the same name, it will get the tasks and dependencies as before because the published version gets the same ID. But you will do a copy automatically also.