And just as a further verification on this, when I had everything working perfectly, I hit the "upgrade" button, to upgrade to 3.2.1, just to SEE if it might work that way.. To my dismay, the problems immediately reappeared, just like before. So now... I will have to uninstall and go back and install the previous version again. But at least I know that was definitely the problem.
This is been a very frustrating experience, trying to fix something for 2 full days that couldn't be fixed in the first place.
About effectively opening myself up to hackers: It's either that, or not using WordPress. I'll keep using WordPress with the versions that do work, and if something happens "hacker-wise", I'll just use something else. That's really the only option I see at this point.