Prove that every two vertices in graph are connected by a path of some length
$O(n/3)$ doesn't mean what you think it means: in particular, $O(n/3)$ denotes exactly the same class of functions as $O(n)$. However, it's not true that the longest distance between two vertices in a $3$-regular graph is roughly $n/3$. Consider the graph

o---o       o       o             o       o---o
|\ / \     /|\     /|\           /|\     / \ /|
| X   o---o | o---o | o-- ... --o | o---o   X |
|/ \ /     \|/     \|/           \|/     \ / \|
o---o       o       o             o       o---o


(where the Xs denote two crossing edges).  The distance from the top-left vertex to the top-right vertex is approximately $3n/4$.