Answer:
Stealthy as a shadow in the dead of night, cunning but affectionate if given a bite. Never owned but often loved. At my sport considered cruel, but that's because you never know me at all.
My first is a creature whose breeding is unclear. My second, a price you must pay. My whole can be found in the river of Time and refers to events of today.
It is a part of us, and then replaced. It escapes out bodies, to a better place. The world becomes its sizeable home. Its passions unrestraint, the planet it roams.
What can you fold but not crease?
What does everyone know how to open but not how to close?
You use it between your head and your toes, the more it works the thinner it grows.
Difficult
Animal
Easy
Food
Funny
Kids
Logic
What am I?