I solved it! I solved it! 😮
Reinstalling did not work, neither did creating a new user account. Skip down to the TL;DR section for what I did.
I created a new user and installed OS X Mavericks again. I didn't re-format the HDD, because I did not want to re-install all of my applications and wait for all that (hours and hours). This did not work at all and in fact, even more stuff "broke." After the reinstall, I could not (and still can't) log in to Apple Support or my Apple ID via Safari (have to use Firefox). Safari is telling me it can't establish a secure connection to the idsma.apple.com or to the appleid.apple.com servers. iMessages and FaceTime were still preventing me from logging in, and everything went to Hades in a handbasket. Oh--I couldn't log in to the App Store, either. I also repaired permissions while booted from the OS install disk--that didn't help either.
However, I DID FINALLY manage to get iMESSAGES TO WORK on my Mac! (This is in my original user account, not the new one I created, just FYI.) I again changed the time zone preferences from automatic to manual (with my correct time zone) and I made a change to the time preferences so that the date is displayed in the menu bar. I did that just to ensure the .plist file was updated by the OS. (I'd done this before, but since this was a new OS, I figured it might help). Then, after reading another user's tip in another thread, I logged completely out of iMessages (and iCloud, just for kicks) on my iPhone and iPad. I also went to Sys Prefs > Users and re-added my Apple ID to my computer's user id/login.
I don't know how much this matters, but when I did that, the "caution" icon (yellow triangle with exclamation point) appeared next to the Apple ID info in the Sys Prefs. I double-clicked it to see if it would tell me anything, and it disappeared. I re-launched iMessages, and I kid you not, it logged me in! WOOT! Then I logged in to iTunes, iCloud, and the App Store. I also re-logged in on my 2 iOS devices, and it's all good. <happy dance>
Now, if I could only figure out the screwy problem with Safari and the idsma.apple.com and appleid.apple.com servers. Meh. I can log in w/ Firefox, and at least I can actually do some damned WORK on my computer now (it's only been 4 days iof this BS)!
TL;DR, try this:
1. Make the time zone on the Mac not automatic (but correct to your time and zone), and change something in that pref. pane so that it overwrites the old prefs file.
2. Log out of iMessages, iCloud, etc., on your iOS devices
3. Un-associate your Apple ID with your computer user login (in System Preferences > Users and Groups).
4. Sign out of iCloud on your Mac.
5. Reboot.
6. Re-associate your Apple ID with your user login. If you see a caution icon next to it, double-click. Should disappear.
7. Launch iMessages and attempt to log in.
8. Then log in to iCloud via System Preferences > iCloud
If that doesn't work, do these steps, then go back to # 1, above.
9. Delete all the Apple tokens and keys in the Keychain (might break some other things--may be why I can't log in via Safari right now).
10. Reinstall Mavericks/OS X.
11. Repeat steps 1–8.
Final thoughts: My serial number did appear in System Information, and I'd never had the motherboard replaced (or anything else, for that matter) on this Mac. However, this is an Apple Certified refurbished Mac, and it could be that the motherboard was replaced before I ever even owned it. Again, my S/N displayed as it was supposed to, but it could that this was part of the problem. Seems far-fetched, but...
