"Once I download something, say a PDF file, can I move it from "downloads" to an icon on the "desktop", or whatever the main screen is on an Android phone? " - Scott
Linda Manager" which is free. When you go into it you can find your file (mine was in Downloads), then press and hold the file to bring up the menu and select "Create Shortcut".
A shortcut to that file is then placed on the next open slot of your pages. You can then move this shortcut around just as you would any app. Press the shortcut and it will open the document. Very cool.