The derivative outputs the slope of the original function for whatever x value you input, so the slope of y(x) at x=-1 is y'(-1). Once you know the slope you can use to get the whole equation of the line.

For the second one, the tangent line is horizontal if and only if the slope is zero, and since the derivative gives you the slope the zeros of the derivative will be the x coordinates for which the slope is zero. Once you have the x coordinates you can plug them into the original function to find their y coordinates.