Answer:
My second is performed by my first, and it is thought a thief by the marks of my whole might be caught.
A king wears one on his head.
It is in every mountain, it's not in any hill, it's not in all the world, and yet it's in the mill.
It covers you, you lose it every day and you'd die without it.
What can go up but never come down?
I have feet on my head
Easy
Animal
Difficult
Food
Funny
Kids
Logic
What am I?