I had simililar problems like yours, but my upgrade scripts timeouted, so I had to run them from console. So, try run them on the console, if you can. The last upgdrade step, must be done using the /install/upgrade.php script.. which timeouted for me, but I guess it checks where it last left and you can safely re-run it. It is important to complete this one. <img src="https://www.ubbcentral.com/boards/images/graemlins/wink.gif" alt="" />
The login problems can be solved if you change your password. The encrypted password format that the admin accepts is now different than the old one (pre 6.2.2.)
