I came across this script that lets you search your site with google in that way <script type="text/javascript"> var domainroot="www.yourdomain.com" function Gsitesearch(curobj){ curobj.q.value="site:"+domainroot+" "+curobj.qfront.value } </script> <form action="https://www.google.com/search" method="get"…
No. You sign up to Google lock in your domain, and results are returned accordingly. I've been doing this for years. It also serves a purpose that your pages are identified in search engines and that it complies with Google's submission criteria.