When a recipe call for active dry yeast and a cup of hot water do u mix the yeast in the water to activate it

The point of active dry yeast was supposed to be that you didn't need to proof it and could just mix it right in with the flour, then add your liquid. I have never done that, because if it doesn't work (foam and bubble) then you've wasted your flour. If you add a tiny bit of sugar and warm water as Diana says, it should foam up and smell lovely in about 5 mins.

But it shouldn't be hot water. 110-130 is about right. 140° water will kill yeast.

Depends on the recipe. Sometimes the yeast is incorporated with other ingredients and the water