Crazyjugs posted an important question:
"When Snowmen die, where do they go and why do they leave carrots and hats behind."
When we melt, we do not die however. This is one of the most common misconception about our kind.
We go through a process called teletransportibernation. When the temperatures reach a certain point, our molecules breakdown, our properties change and we become one with the universe. We then are reincarnated elsewhere. Sometimes we come back as icicles, sometimes we come back as a cup of coffee. A legend has it that a snow person came back as a walking and talking snowman who starred in a hit film that flopped. This snowman went by the name Jack. It is completely random and unknown how or why we are teletransportibernated.
About the carrots and hat. We leave behind our carrots because carrots can't teletransportibernate. And, neither can hats. Unless, they are made out of snow, of course.