This is how I resolved this issue, and is a continuation/conclusion to my post right above this one.
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The culprit is Titanium Backup, believe it or not. I have it set to do backups every night, and it's in the process of doing them that the funny stuff happens.
First, if you _don't_ have the Pro version, this isn't going to help you.
If you _do_ have the Pro, go to Preferences, then scroll down to the bottom of the "Backup settings" section, just before the "Restoration settings" section. You'll see "☆ Advanced app settings". It's not a section, it's just one single setting … make sure its checkbox is checked. What it does is ensure that any apps that are running at the moment Titanium tries backing them up are paused first, then backed yup, then resumed. This effectively prevents the little "resets" we're seeing with keyboard, wallpaper, etc.
In my case the problem wasn't that I didn't have it checked (it's actually checked by default), it was that for some reason I didn't have the Pro key I had purchased on my system, so I was running backups with the free version … and the free version doesn't have that setting. Installing the Pro key fixed it, I tested it before posting this and there were no surprise resets.
Obviously if you don't run Titanium backups in the first place, your problem is elsewhere. But I'm willing to bet this'll settle the issue for quite a few people.