You shouldn't be guessing anything, but yes, h = 2 when t = 0.
Your answer to part b) is correct.
As for c), I would think about the symmetry of the quadratic, which means that everywhere except the turning point, there are always two t-values for every h value, and due to the symmetry, the turning point has to be located exactly halfway between them.
You know that when t = 0, h = 2. Can you find another t-value for when h = 2? When you have that, average them and that gives you the t-value of the turning point. When you have the t-value of the turning point, substitute it into the equation to find the h-value of the turning point.