I do not believe that you can open the authentication page in a separate window according to the terms of Facebook (I worked for Zynga, and then we could not do this, so I do not know how you will achieve this now legally).
Secondly, you look back in QWebkit, I believe. From a user interface point of view, this should provide access to and interact with websites. If you need a good page with all the bells and whistles, it hurts me to say this ~, but use ActionScript 3 or Ajax and this bundle.
This post would probably respond better if tagged in javascript, php, and as3 sections. Since the php guys will know the right hacks to get your intended result, if possible