OK, let's take a step back here.
Starting at 10.1, ArcGIS Server is a Java application that has web server components.
It does not install to IIS, so you wouldn't see an ArcGIS Application there in IIS quite yet.
There *is* a Web Adaptor component that can install to either IIS or to a J2EE server (sounds like you'll be using IIS).
When you install the web adaptor, you'll see that application installed into IIS.
Background: The web adaptor essentially acts as a proxy from your web server front end to the GIS Server back end - essentially, it proxies requests into https://yourwebserver.com:443 to the GIS Server at https://yourGISServer.com:6443 (or from https://yourwebserver.com to http://yourGISServer.com:6080)....
At this point, it's expected to not see an application for ArcGIS in IIS. Instead, you're running the GIS Server on HTTP ports - outside of IIS - , which is expected.
Please copy and paste the following into a browser running locally on the machine that the GIS Server was installed on:
http://localhost:6080/arcgis/manager
By default, assuming everything installed correctly (which we're not sure about yet), you should see a page that asks you to create a new ArcGIS Server site.