What started this oven-fire - The ingredients or the oven? Should I use a different burner or..?

Milk does boil over very easily, so when it and the butter (grease) hit the burner, that's when it caught into flames. Just clean up the burner after it cools down and you'll be fine. Don't tell maintenance or your landlord will know, and the less problems you have with your landlord, the better.