Longest repeated substring
Yes, the number of nodes in the suffix tree is $O(n)$.

Here's how you could infer that.  The suffix tree can be constructed in $O(n)$ time.  There's no way to construct a tree with strictly more nodes than that, since it takes at least $\Omega(1)$ time per node you construct; so from the fact that the running time of the suffix tree algorithm is $O(n)$, you can conclude that its output must have size $O(n)$, i.e., the suffix tree must have $O(n)$ nodes.

Here's another way you could have inferred that.  There are $n$ leaves.  Also, each internal node has at least two children.  In such a tree, the number of internal nodes is at most the number of leaves, so in this case, there are at most $n$ internal nodes and $n$ leaves.  It follows that the number of nodes in the tree is at most $2n = O(n)$.

This information is actually found in the Wikipedia page on suffix trees, which explains that any suffix tree has at most $2n$ nodes.  In the future, before asking a question here, please check standard resources, such as Wikipedia and a textbook.

It follows that the tree can be traversed in $O(n)$ time.