If you install an application from the Office Store, let's just say Word for example. When I then re-open word, can that app / plugin run a background task or must the app be engaged with by the user?
The App always has to be opened. It is essentially nothing more than a web page, and code doesn't run at all until the web page in your task pane is opened. The user always has to explicitly launch the add-in; there's currently no provisions for automatically launching an add-in when the host application starts.