That wouldn't work because Google would not return the proper content, since the host header would not match google.com. HTTPS would not work either, like you s

That wouldn't work because Google would not return the proper content, since the host header would not match google.com. HTTPS would not work either, like you said, because the cert would not match the new domain. If they didn't check host headers, then no, the browser would not send the cookies for google.com, since the browser has no idea about your DNS changes and is only looking at the domain name itself.