If you are going to ride main streets, you should not worry about the tires. At least in Warsaw most streets are salted such strongly, that they are white from salt, dry and clean. -10 is not a problem for "summer" bicycle tire on dry road. But if there are bicycle paths, even along those neatly cleaned streets, keep in mind that it may be icy and slippery the whole winter.
Be careful during thaw. Give up commuting with trailer when bicycle paths are covered with thin ice surface.
Make sure, that trailer hook is safe in every possible fall over scenario of your bicycle (turns, etc.) and it will not pull the trailer and make it fall over also.
Keep in mind, that your pedaling helps you keep warm. Your sons should be dressed even warmer than for winter walk. Try work out some mini-excercises for them to make their muscles work, but will not affect trailer stability.