Most likely the problem with the livesite...
You look in your site configuration (General Configuration, select the Server tab) you will see that the livesite is given. Usually it is either www.mydomain.com or mydomain.com.
One way to deal with this is to put a redirect from, for example, mydomain.com to www.mydomain.com so that you and your users always login from the correct url. You can do this with .htaccess.