Sadly, I can't pinpoint when the problem started because I don't recall how long it's been since I tried to successfully do the things that are resulting in a 404.
Things like: When viewing the Spam comments, I can "delete permanently" on each individual comment but I cannot select the whole page of comments, nor can I just click on "empty spam". The latter two result in a 404.
I can't delete a plugin, I get a 404. I tried to re-update WordPress, wondering if it was a botched update. It does the update but when it's time to display the "The update was successful, here's what we changed" page (about.php) it 404s. I looked, and the about.php IS in my file structure.
I don't get a 404 from anything other than Dashboard admin stuff, so I can't even show you what the problem is.
Things I've done:
*Deactivated every, single stinking plugin.
*I use GoDaddy for hosting, so I called them to make sure there wasn't an error on their end that they could detect. Using Godaddy, I can look at the File Manager and look at the files in "history" view; I can revert the files to how they were on any day in the last month. That's a good trick for if I get hacked, to quickly fix the site. The revert didn't work, I even reverted to prior to the 3.5.1 update.
*Activated the default WP theme instead of my usual theme.
So it doesn't seem to be the host; it doesn't seem to be a plugin, or the theme, or a botched WP update. It doesn't seem to be a hack. I'm out of ideas.