What I want to know is whether it's possible for a web developer to use some kind of technology which hides the actual URLs of a site and presents to them cleaner alternatives. Could a website for example have a web page which links in reality to sitename/fmi/webd but present the URL to the user as being sitename/differentlocation?
Use apache redirection commands to achieve that. http://httpd.apache.org/docs/current/rewrite/remapping.html
You could put it in a frame. But why hide the URL? Or if you want it to be compatible with a company's url, then for "companyURL.com", have DNS assign your FileMaker Server machine to something like "fms.companyURL.com" so that the company's URL is still there and the machine name is just appended to the front of your company URL. That way connections to it appear to all be apart of the same system.