A static IP will point to your router 24/7. From there, what you do with it is down to you - you can open up ports and direct them to a specific machine on your network (eg running a webserver) or you could host a chat / email / ftp server etc.. It's up to you.
But in short, yes, you can host a webserver on your IIS machine and direct external traffic to if via the IP address. You may want to put that machine into a demilitarised zone if your router supports it.
I've run a webserver on my personal PC for years with no known issues from hacks etc BUT you have to be careful if you're running any php / perl code (or any other webserver language) as it could be abused if you don't know what you're doing so it may be best to stick to html pages for the time being.
I need a new signature... i'm bored of the old one!