Does commerce send an order notification email to site admin (site email) out of the box?
It doesn't look like it does, no. There is, however, a rule set up to send the order notification to the user. It's called "Send an order notification e-mail".

You can easily clone this rule, but instead of sending the email to the owner of the order, send it to the admin's email address.

The admin user isn't available as a data selector for this rule, so you'll either have to expose it yourself somehow (not sure exactly how you'd do that without custom coding off the top of my head), or simply switch to 'direct input mode' and put the email address in manually.