Wherever I put my cursor in any DW page, design, split, or code presentation, a strange little box appears near the cursor and only goes away when I click the cursor in another place on the page and this it reappears within about 2 seconds.
I had trouble taking a screen shot of it. This was the best I could get because it quickly disappears when you setup for the screenshot.
It's the Code Navigator. If you click it, you'll get a list that allows you to inspect, and navigate to the CSS of the current location within your page.
Open it, and you'll see the checkbox at the bottom right that lets you disable it.
Once disabled, the icon won't show up again, but you can activate the Code navigator in Windows by holding down Alt and clicking. On a Mac, hold down Opt + Cmd and click.
What's your platform?
If Windows, It sounds like you have turned on the Magnifier.
Open Control Panel / Ease of access center.
Click Make the computer easier to see.
Scroll down and remove the check mark from the Turn on Magnifier option.