We do not recommend using the webdisk, I'd recommend using FTP or cPanel FileManager. You could use a free ftp client such as FileZilla on more than one computer. You can use the webdisk on more than one computer if you want.
I would recommend doing the upgrades via PHPBB not Fantastico. From what I see it looks like your forum was updated via PHPBB and not Fantastico, leaving Fantastico behind.
Free accounts do not get SSH Access, You would only get SSH access if you we're a paid user and had a VERY good reason. If you are needing SSH Access, I'd recommend you purchase a VPS at: https://clients.x10hosting.com/cart.php
It looks like your account has been moved to the new server, You may need to wait a few days before your PHP is set to Advanced. I'm trying to contact the person in charge of Stoli's server transfer.
The name servers are the same. The IPs are listed in the server transfer threads. You may need to wait a few days before the PHP is correctly set to your level.