Nah - no domains need be purchased and no static IP address is required. Check out
http://www.dyndns.org. Read the FAQs there. It will do exactly what you want.
I've never heard of Aprelium's Abyss Web Server X1; but why not just use IIS? Or if you want an extremely responsive efficient webserver that is deployed on the majority of websites on the internet (and is free) go for Apache. Apache configuration is done through text files though and not everyone is comfortable with that.
DO NOT run a web server unless you either a. have a well configured firewall or b. don't care what people do to your computer or what they use it for (e.g. you are deliberately creating a honeypot).
The bandwidth comments in the last post are not necessarily correct. Here in the UK at least, most broadband has wider bandwidth downstream (traffic to your computer) than upstream (traffic away from your computer. e.g. I have 2Mb downstream - brilliant for browsing the web, but 128k upstream - useless for hosting high volume websites (all the users would experience very slow response times).
Another thing to consider: it is not advisable to run a production service (web, ftp, news, gaming, IM, etc) on the same computer you use for your work. Otherwise you may well find that conflicts occur or that performance (for you or for your users) is unacceptable. Epsecially don't even think about providing a web service on a computer that you regularly use for gaming.