I found out recently that the wooden plug was put there to strengthen the steerer/fork. Leave it there. I took mine out when I painted my frame and now I' fretting about putting it back with the hole lined up for the brake. For the stuck stem, try PB blaster and let it soak. If the headset is not stiff or gritty, and the reach is ok for now, ride it as is while you contemplate more drastic measures. All the while, try the stem every few days/ weeks. Cool bike. Lucky for you and your kids. Best wall hanger ever if your kids don't ride.

Jeff