Elgg itself is free, however if you wish to install it through Softaculous in Webuzo you need a Monthly/Yearly Webuzo VPS license. Its $2.5 a month or $25 a year.
If you wish to skip Softaculous, you can go download Elgg from there website and set it up yourself.
https://elgg.org/index.php
As for domain names, some vps providors such as Azure provide you one when you create the Virtual Machine an example: http://mysite.cloudapp.net
Otherwise you can access it through the VPS IP address.