You take an old gum machine, paint it up, place coca cola decals on it and sell it. Without getting written approval, is this trademark infringement?

The answer is "it depends." The purpose of trademark law is to avoid consumer confusion regarding the origin or sponsorship of goods. It could be argued that no reasonable consumer could be confused into believing that the Coca Cola Co. endorses or sponsors these gum machines. Is it a good idea? No, it's not.