All depends on what you're trying to do, and the system requirements your software has.
Technically you can run any intel based software / OS on a PC that meets specs. Server class hardware defines a level of performance, redundancy, fault tolerance, and if you go the OEM route, troubleshooting and support tools.
If you're hosting your site for commercial purposes, you need to make sure your ISP does not require you to open a business account with them, most do.
You'd want to have a static IP address, but I believe if you register with ddns.org, it will track your dynamic IP... dynamically.