Automorphism group of the Alternating Group - a proof

For the first point you have to show that the three cycle acts trivially on the orbit of the subgroup $A_{n-4}$ of its centralizer that has size at most $n-4$. Since primitive permutation groups (other than those of prime order) have trivial centralizers in the symmetric group, that would follow if you could show that the action on that orbit was primitive. But it must, be because otherwise its point stabilizer would be properly contained in a larger proper subgroup, which would necessarily have index less than $n-4$ in $A_{n-4}$.

For the second point, the only essentially distinct images for the two $3$-cycles are (i) $(1,2,3),(1,3,2)$; (ii) $(1,2,3),(1,2,4)$; (iii) $(1,2,3),(3,4,5)$; and (iv) $(1,2,3),(4,5,6)$, and they only generate a subgroup isomorphic to $A_4$ in Case (ii).