Of course they didn't just live in the artic. Because during the ice age period almost 25% of the earth was covered in snow and ice.
Mammoths went extinct in large part because of our human ancestors. To say animals should stay extinct is stupid.
Moreover, if they do successfully breed them, they won't be exact replicas of the originals. They will basically be cold hardy elephants. So why wouldn't they be able to coexist with other animals. Today's elephants don't go around squashing every other creature they come across.