Good call on the key. This is exactly why the mower wanted to stall in reverse. This is my first riding mower and I should have spent more time learning about such "features". So, after I screwed up and ran it in reverse without turning the key to the correct position, it caused a backfire, and since then it has been discharging white smoke and running roughly. Not super rough, but something is clearly wrong.
I just started it, ran it for a couple of minutes, and checked the plugs. Neither were wet with oil, but one had more black buildup on it than the other, although both were quite dark. Those plugs had about 30 minutes of run time on them.