Just to be clear, if you navigate to ~/Applications, or /Applications, wherever Pages is installed, and click on it there, you get a Desktop folder? I would suggest that maybe you can do a "Get Info" on Pages in the location where it is installed and see what it says about that icon.
Bear in mind that if you open Pages normally, a Finder-like window opens up to show potential documents. I think it defaults to where you last opened Pages, but I'm not sure. There is a "New document" button in the lower left corner of that window. Is that what you are seeing?