There is no listener to capture when the browser is launched.
I imagine that you might implement your requirement by having a background program which polls the "visible applications" list from ApplicationManager. You could then push a global-modal dialog when the browser shows up in the list.
You can find out which Field has focus, get the UiApplication context for the Field and find out the application that owns the execution thread associated with the field. It is not always the application the field 'belongs to', but it usually is. Sounds like a lot of work for something so dubious.
The Browser is always on because it listens for push events.
My other Blackberry is a PlayBook.