OK, I wouldn't worry about upgrade-functions.php at the moment. It won't be in WordPress 3.7.1, so what you have is fine.
The only thing I've seen in this thread (and the previous one) that makes any sense is the "headers already sent" error, as that would prevent the test cookies from being written.
As far as the cause of that error (headers already sent), it's not necessarily wp-config.php, but often is. But it could be any php file with extra white space after the closing php tag at the end of the file. (which is why the closing php tag at the end of a file is often not used)
Once again, if it were me, I'd either delete and re-upload core files or I would just back everything up (I'd do that anyway) and do a complete fresh install (using your current wp-config.php file or copying the content over to the new one so you can use your existing database) and then bring over theme and plugins (from your old wp-content directory) over an item at a time. I'd probably go with deleting and re-uploading core files first, making sure FTP is set to automatic. If that didn't work, then doing the complete re-install (WordPress only, not the database) might be the only way. Get a vanilla install working and then bring back in your theme and plugins. It looks like over the course of the last two or three days, you've made a lot of changes trying to fix things. Sometimes it might be best to start at square one and go from there.
You definitely need to eliminate the "headers already sent" error.
As far as a corrupt file from the upgrade, I just meant originally. That happens sometimes. Anyway, I'm not sure I can offer much more advice, but I don't mind answering (or trying to :) any other questions. I don't think WordPress itself is the problem, but more likely something to do with the install or a theme/plugin compatibility issue.