Answer:
Runs smoother than any rhyme, loves to fall but cannot climb.
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.
If you happen to wear one of these warm garments around, you might get doused in red paint.
You use it between your head and your toes, the more it works the thinner it grows.
I am never quite what I appear to be. Straight-forward I seem, but it's only skin deep. For mystery most often lies beneath my simple speech. Sharpen your wits, open your eyes, look beyond my exteriors, read me backwards, forwards, upside down. Think and answer the question...
What did the piece of wood say when he saw the screwdriver and screws approaching?