Tonight... That's more than I can say for the past couple of days!
We'll see how long for. More extensive tests tomorrow. I cleaned off the spark plug again, and we emptied out the fuel (which looked like it had very little to no oil in it) and replaced it with a fresh mix. (which is really something we should have done earlier) It started right up twice in a row for two rides around the block without stalling.
It is a General 5 Star ST with a Minarelli motor that takes a 40:1 mix. It's currently a new Champion L86C spark plug. I checked it after my two laps and it was dirty again, but not wet and covered in grime this time.
There aren't any modifications and the choke isn't sticking, it's definitely popping up when I give it full throttle. I really don't know about compression, though.
If I have problems with it again tomorrow, I'll definitely have to check the rest of those things that you two suggested. Thanks!