when I import a photo to the Organizer - Do I have to keep a copy on the desktop

When you import a photo file into the Organizer it does not create a new copy of the file.  It merely creates a database of your files wherever they are located.  So, when you delete a file outside of the Organizer e.g. through a Windows File Explorer window, the file still exists in the database, but because you deleted it, the Organizer can't find it. 

When you edit an image and save it, there are several choices that can be made:

If you truly want to delete the original file after editing it in the Editor, the easiest way would probably be to overwrite the original file by using the same file name to save the copy.  When you open a file from the Organizer into the Editor, it automatically gives a new name to the file that has been opened - with "edited" as a suffix.  You can change this back to the original name and the original will be overwritten when you click the Save button.

The normal practice for most users is probably to save the edited copy  in the Organizer and to Save in a "version set" with the original.  When you do this, the Organizer creates a "stack" of the original and edited copy, so that only one thumbnail is seen in the Organizer grid.  You can then view both versions of the image by clicking on an arrow to spread the stack out.

In this screenshot, you see:  1.  Icon in top right corner of thumbnail, showing this is a stacked version set; 2.  An arrow to click and see all photos in the stack; 3.  The original image in a complete version set; 4. The edited copy of the image; 5.  The arrow to close the stack.

If you are going to use the Organizer, you should conduct all file management operations within the Organizer.  If you select one or more images to delete in the Organizer, a message will pop up when you hit the delete button asking whether you just want to remove the file from the Organizer or also from your hard drive.  This will also avoid the problem you are having.

Hope I haven't confused you too much.