This week's book giveaway is in the OCAJP 8 forum. We're giving away four copies of OCA Java SE 8 Programmer I Study Guide and have Edward Finegan & Robert Liguori on-line! See this thread for details.
You have things back-to-front. The domain name is used to locate the server, not vice versa. When you send a request to http://www.mydomain.com, what you are actually doing is making a request to Domain Name Services to lookup the address of the machine named (or aliased) "www" in the domain "mydomain.com". Then, for good measure, since you requested the http protocol, the message is sent to port 80 on that machine. Tomcat conventionally uses port 8080 (or, apparently, port 8180 on the rpm distribution of Catalina - countrary to what the documentation says). You could just as easily make it run on port 80 by changing the config file (server.xml) - if you have low-port privileges, but more commonly a forwarding/proxy server is used, such as Apache with mod_jk or mod_webapp.
An IDE is no substitute for an Intelligent Developer.
He still needs DNS, though for the host part of the server to resolve properly and Tomcat would have to be configured to use port 80. The redirect may be doing it the hard way. I'd hope that like Apache, Tomcat would have a config parameter that allows you to name the default page anything you like. Failing that, I'd look to see if the mapping facility would save a redirect. I know I can map index.html to a servlet - I don't know if a JSP can be mapped, too, since I'm pulling all this stuff from the top of my empty little head.