The free version isn't damaged. That's OSX telling you that because the program was built before OSX signing existed. There's a setting that you can turn off in OSX that will allow you to play the game without it lying to you about being "damaged".
The online version won't work in Chrome anymore since Chrome disabled plugins. There's a way you can get around this for the moment in Chrome, but the easiest thing to do is use another browser to play the web version.
There's also the HTML5 version of the game which should work fine in Chrome:
http://www.DesktopDungeons.net/HTML5