I have an app, which interacts with a foreign webserver which sets a session cookie upon login. Certain pages I would like to display in a webbrowser, instead of processing the HttpGet request. Unfortunately the website uses frames which is supported very poorly by Android's WebView. But it works fine with Opera Mobile Browser.
Is there any way I can pass the session cookie from my App to Opera Mobile?
There is no way.
Opera is an external Android app and all you can give for it is an URL. This URL must contain login parameters. WebView is an UI component which gives you much greater control over it.
If you cannot change the web server to support login-by-magical-URL then you are probably out of the luck.