Is it possible for a java app to open a browser and to be able to leave that browser open running in the background so that you can go back to the phones normall functions?
As far as I know, in J2ME MIDP 1.0, you cannot do this.
I believe that in MIDP 1.0, you cannot launch other applications, but with MIDP 2.0, midlets can launch other applications (you might want to double check me). But, I'm doubtful if MIDlets would be allowed to launch the mobile phone's native browser.
However, Symbian applications usually have more access to native phone funcationality, so I would check to see if Symbian Apps could do something like this... or lend functionality from other native phone applications.
For better answers to your questions, try posting these kind of questions in the discussion board related to the programming language like... the Forum Nokia Java Discussion Boards
If you have a questions about WAP/Browsing standards or browser specifications, then, this Browsing discussion group is the place.