Can you use a PC as a server, the answer is yes.
Check your Internet Service Provicer contract to see if it allows you to legally run a server on the connection you want to use. On this I'm not going to go into other than to say that if your contract does not allow you to run a server the ISP may take you to court or just turn off your access.
As for your Dynamic/Static with or without router question, the simple answer is that it can be done provided you register with a Dynamic DNS provider and run a small program on your computer that keeps things updated.
The following link will give you much more information about Dynamic DNS setups than I care to think about.