Can the gross ever give the subtle anything? That is, can the soul become stronger by the right use of the body such as exercise, right food, and so on?

Certainly. The gross can and must help the subtle. The body is gross, but in its sound and perfect condition, it helps the mind and the subtle existence considerably. One cannot, of course, make the soul stronger merely by taking vigorous physical exercises or by eating judiciously. But if the body, that is, the physical consciousness, aspires to grow in the Light of the soul and tries to fulfil the Divine in the physical itself, then the progress of the soul becomes easier, faster and more fulfilling.