it would appear that for any thrown object has a height that can be expressed using a quadratic expression. The maximum height occurred in the middle of the two zero values. Will this always be the case? What if the ball were thrown from a platform of height 1 m?

Answer 1

of course. All parabolas have the vertex midway between the roots. The quadratic formula makes this easy to see, since the roots are at x = -b/2a ± (some junk)

Starting from a nonzero height just places the smaller root at negative value.