Is there an alternative to the five-finger swipe out gesture, allowing me to copy things to/from the desktop?
I would suggest you set up a hot corner to show the desktop (this is what I do):


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


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


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