Thanks for suffering the pain for all this - your pain likely saved me many hours of trying to figure out what was going on when I had the same problem!
Note that, rather than totally changing to "Allow applications from anywhere", you can just allow an exception for OpenOffice (copied from somewhere in Mac's Help Center):
- In the Finder, locate the app you want to open. Most apps can be found in the Applications folder.
- Press the Control-key and click the app icon.
- Choose Open from the pop-up menu.
- Click Open.
The app will be stored as an exception to your security settings, and you will be able to open it in the future by double-clicking it.
I did it this way, although to be honest the whole "unidentified developer" thing is proving so annoying I may soon switch it off anyway.