How to show an application only in the dock assigned to a desktop

I would suggest you set up a hot corner to show the desktop (this is what I do):

1. Open System Preferences
2. Click Mission Control
3. Click the "Hot Corners..." button at the bottom
4. Set the corner you want to use for "Desktop" (I use the top right)
5. Click OK

Now you can move a file from a Finder window to the desktop with just the mouse:

1. Click and drag the file you want to move to the hot corner you set up, which will show the desktop
2. Drop the file on the desktop

In the terminal, do this:

defaults write com.apple.finder CreateDesktop -bool false && killall Finder

After this, there'll be nothing on your Desktop (but still in ~/Desktop, i.e. your Desktop Folder). It'll look like this:

Beautiful, isn't it? If you want to change it back:

defaults write com.apple.finder CreateDesktop -bool true && killall Finder