You have many questions

Let me try to shed some light.
First does HU offer SSH access? If so backing up and moving will be a simple task. If not it can still be simple, but a little more complicated time consuming.
First get your database backup performed. You can do a full export if its small enough from PhpMyAdmin, download it to your computer.
Next access your ASO account via the IP address, create your database, and restore that database you just exported.
Next move download all your files from your old account. And simply reupload to the new account.
Then change your nameserver information to point to the ASO set.
Its pretty simple, it just sounds complicated.
And if you put password protection on one copy of the vBulletin so its not accessible to the public, then you can run the two at the same time.