Deepest place is the Mariana Trench at 36,070 feet below sea level
Highest place depends on your definition
Mt. Everest is 29,035 feet above sea level but it sits lower on the equatorial bulge than does Ecuador's Mount Chimborazo in the Andes which is 20,549 feet above sea level but sits higher on that bulge so that it is farthest away from the Earth's core...closer to the moon than Everest.