Can I prove all implication proofs like $A \to A$ or $A \to B \to A$ in both Sequent Calculus and Natural Deduction or just in one of them? So for $A \to A$ can I use the right implication rule in sequent calculus to prove it?
Of course this depends on the proof rules in the two systems. But the usual formulations of sequent calculus and natural deduction for classical propositional logic are equivalent: $\varphi$ is provable with a natural deduction proof with hypotheses $\Gamma$ iff the sequent $\Gamma \vdash \varphi$ is provable.
Proving $\vdash A \to A$ in sequent calculus is almost trivial. We have $A \vdash A$, and now use implication introduction.