Yes, you can have more than one WordPress installation on a domain, either in separate folders or subdomains. Each can use the same database (with a different prefix) or each can have its own.
You need a copy of Genesis and a child theme for each installation.
Alternatively, you can use the multisite feature of WordPress where you add a line of code in your wp-config.php file and then you can add multiple sites/blogs to one single installation (with one Genesis and multiple child themes), all accessible from one master account, and each individually accessible by its users, and to the outside world, separate sites with separate themes, plugins etc.
Multisite can be set up to use subfolders or subdomains and it is possible to point individual domain names to separate multisite installations.