Buy your new domain, log into your registrar and point the domain to your nameservers. Log into your cpanel and add the domain, make sure it points to the folder that contains the files of the old website.
well true, he can forward DNS to a new name but it would be best to make the changes at the domain registrar and have them create a new account or create your add-on.
You can change, but you'll loose the traffic initially, but the traffic from previous domain can be redirected to new domain by using 301 redirect.