As an engineering student, is it very important to understand and memorize all the proofs of theorems in Calculus 1 ?

Understand = definitely

Memorize = very useful (especially for exams), however, very few people manage to memorize ALL theorems. That is why someone invented books. Mind you, as a young student, it is usually easier to memorize tons of things than if you wait until your are an old engineer... when you tend to forget stuff, like where you put the car keys.

Yes