Control Panel> Accesibility Opitions> Keyboard tab.
See if the boxes are checked and uncheck them if they are.

You fingered the likely culprit first, and profiled it with the damning description “wireless”. The smoking gun was “occasionally something has landed on a keyboard and stuck a key”. The OS theory was a red herring, of course. The forensic evidence was the behavior of the cursor, moving two steps like a knight in chess, for example, when you typed two characters. Just as if you’d typed ctl-T ctl-H. It just so happens that ctl-H is the standard way to produce the backspace character. Can’t think of an equivalent for ctl-T, though.

You might have a stuck ctl key, either mechanically, or in the keyboard electronics. I have one of those wireless keyboards in the living room, and have had problems of this sort. Try turning power off and on (jiggle the handle). Check the transmission path between keyboard and computer; if there’s a USB dongle, make sure it’s in the open. Check the keyboard batteries; the long lifetime of mine makes me complacent. When you’re frustrated and driven to distraction, bang it on a carpeted floor key side down, to see if you can unstick it or dislodge obstructing crap. If that doesn’t work, repeat the procedure on concrete. Sweep up the pieces and dispose responsibly.

Then wonder if maybe alcaray and TB weren’t right after all.