Yes you definitely need concepts from calculus to compute the total impulse transferred when the contact force is variable. Though calculus per see, is also basics even for a physicist. As physics uses the langauge of mathematics, it's hard to get it without math basics such as calculus and some linear algebra.
My main area is theoretical physics and my own favourite area is the foundations of the interactions, such as quantum mechanics and relativity. Basically to seek deeper understanding of why the laws of nature are what they are, and how they were unified at the big bang, mainly from an information theoretic perspective of inference.
Yes, a "push" would have very LOW peak force, and can MOVE the subject by not damage it. So ideall you need both high force and maintain it trough the target. So both concepts matters.
True, high speed and high energy are not in contraditiction. A good technique gives a good amount of both, but with a poor technique you can get one without the other. You need to think about both and balance the technqiue. In principle differnt target require different balance. Typically to break brittle brick, you may need high force, but not very much energy. But to break something that is not brittle but deforms, you may need more energy by maybe less force, and more drive. To break a brittle concrete brick does not required much contact time, as once the peak force is there it just cracks. To drive through someones ribs would need much more drive.