Why contraction and weakening rules are the upside down?
Contraction is called contraction because it contracts the two hypotheses into one: $A, A$ is contracted into $A$. Weakening is called weakening because it allows one to deduce a weaker fact: $\Gamma \vdash B$ is stronger than $\Gamma, A \vdash B$ since fewer hypotheses are needed to obtain the conclusion. The names come from forward reasoning: deducing facts from other facts. When you're deducing new truths, you can take already-proven truths and add hypotheses or remove duplicated hypotheses.

When you look at contraction from the point of few of backward reasoning, i.e. examining why a fact might be true, then the contraction duplicates a hypothesis. Similarly the weakening rule discards a hypothesis. When you're trying to prove a goal, you can duplicate or discard hypotheses to get a new goal whose truth implies the truth of the original goal.