I am using Firefox Developer Edition and when I had the Inspector open, which is accessed by right-clicking an item and selecting Inspect Element, I then right clicked something in the Inspector and selected Show DOM Properties, which you can see in the screenshot below:
However I cannot figure out for the life of me how to close the DOM Properties window.
How can one do this?
The feature you're referring to is called the "split console." You have it toggled on currently. It will show when a tab other than the console is selected. You can click in the split console, or command+alt+k on OSX, and hit escape to close it, or click on the console tab, or click the button in the top right of the dev tools to toggle it back off. I find it most usefull with the debugger tab. When the debugger is paused, you can access variables within the scope of the breakpoint.