Can I edit domain settings if I didn’t create the site?

Sounds like this is a Multisite installation, and if so — he needs to add you as a Network Administrator not just a single site administrator.

I need to be able to add a sub-domain that points to a different website

What exactly do you mean by this? You want to add a subdomain that points to a different server? A different WordPress installation? A different site on the same WordPress installation?

If you’re just trying to setup a subdomain to point to a different site (different from the main WordPress one) — that needs to be done through your friends web hosting control panel — not through WordPress.

If it is a multisite and you’re trying to add a new site (setup as subdomain), as mentioned, you need Network Administrator access not just standard Administrator