Usually the rack servers are better since they were designed specifically for being a server.
And yes, it is possible to run a webserver and a game server at the same time. For a Minecraft server the specs of the server you linked to will do just fine.
If you're using Windows, the easiest solution is WAMP, but I'd set a root password if the webserver is going to be public on the Internet. Most people go with a customized Apache setup however.
I'm probably going to use linux for the server, its more stable than windows. But I don't know a lot about security of servers and Linux so I'll have to look into that some more then. Thanks anyway!
This is a decent primer to setting up LAMP in linux. It doesn't go over DNS or domain registering at all, but it's enough to get apache up and running locally. Should give you a basic idea of what you are getting into. http://www.computerforum.com/199498-lamp-server-setup-part-1-basics.html