It sounds like file system permission problems where the thread that is executing does not have permission on some files so it prompts for a different user (maybe you have basic or digest authentication enabled, I think otherwise those requests would just fail with access denied).
The user that needs permissions is the user that is the identity on the app pool.
You upgraded from a very old version that ran under .NET 2, hopefully you went all the way to .NET 4. Possibly your host had to physically move your site to a different machine to give you a newer version of .NET or by changing to a .NET 4 app pool it may have a different identity. Moving a site can result in some funky file permissions, you may want to review the notes in the article Moving an Installation of mojoPortal to a Different Machine, there are some old temporary files that you could delete that may help.
Other than that you should try to reset the file system permissions for /Data and /App_Data, clearly if nothing is being logged then the log file is not writable. You could delete the currentlog.config file and if folder permissions are right it will come back on its own. Probably best to remove permissions first then re-add them and make sure it is set to propagate permissions to all child folders and files.
Hope that helps,