Sure. To run servlets/JSP with an IIS server you will also need a Servlet Container which plugs in to IIS. Resin (URL above) and JRun are both popular Servlet Containers which plug in to IIS. Or you can run another servlet server alongside IIS on a different port (eg. 8080 instead of the default 80).
When you come to put your systems on the real internet you will either need a dedicated connection to your machine(s), a dedicated machine in someone's network center, or space on someone's servlet-capable hosting service. All of the above are possible at differing prices.
If cost is an issue, the cheapest by far is the third solution. If you plan to go this route,
you should seriously consider getting the same software setup as the hosting provider, so you can do real tests on your local network before publishing your work to the internet at large.
Also bear in mind that most low-cost hosting providers use Linux (they can be cheaper because they don't have to pay Uncle Bill). Indeed the majority of servers on the internet run Linux, so if you are really keen on setting up your own mini-internet, you should consider having at least one machine running Linux so you can become familiar with its operation.