NO, unfortunately a home server doesnt do that, nor does any other server. The only reason you would want to host yourself if if you had a dedicated server that can handle the load, or its for testing/low traffic purposes. To get a domain name of any sort you have to pay unless you want a country level domain. eg. www.yourname.tk. Also as Skemcin said you probably have a static IP, so it would also be difficult.