Well, at the end of the day, it seems as though the DMG that is getting downloaded is corrupt for one reason or another. Perhaps a problem with your ISP and it's becoming corrupt during transit, or some other anomaly.
I suppose you could try allowing all install sources and see if it makes a difference, but it shouldn't, based on the message you're receiving. To do this open System Preferences > Security & Privacy > General. You'll probably have to unlock this pane with your admin password. Select 'Anywhere' under 'Allow applications downloaded from'. Try again.
Worst case scenario, do an OS reinstall from the recovery partition. This should only over-write the OS and leave your data and Applications in tact (unless you use Disk Utility and erase the OS partition manually before reinstalling). After this, try updating again.
Just some thoughts of things to try, short of taking it to an Apple store. Although, the latter might be a better thing to do if you don't want to spend too much more time on this.