It's possible, but it will take a bit of work. The easiest place to start is AppServ (
http://www.appservnetwork.com). It's a Windows based Apache installer that includes just about everything you will need for running a server.
As far as a domain, the best thing you can do is use ZoneEdit (
Free DNS service - Easy, web-based domain manager - ZoneEdit.com) with DirectUpdate (
http://www.directupdate.net).
If you don't have a full domain but just want a short and easy to remember address, use No-IP (
http://www.no-ip.com).
That should be enough to get your feet wet. If you get stuck anywhere, just ask. Good luck!