My name sounds like a child's toy on a shelf, Joined with a fish's tool for steering itself. I live in a school but I never pass tests, I click and I whistle with air in my chest. What am I?

Yesterday’s Tomorrow is where you will find me. Time will tell.

A soup, spell, On hearth or in oven, A vessel for warming, With family and coven.

If it’s after an ‘F’ it’s the one-hoof-lung. If it’s after an ‘S’ it’s the one-whos-lung. As itself it’s a taste that won’t get off your tongue. Overstaying a welcome. Hanging after thing’s done.

I live on a horse. You wouldn't believe me if I told you that the next major improvement came almost one thousand years after my invention. And, oddly enough, I came thousands of years after the invention of another.