The typical operations on a tree are insertion, deletion and traversal - including successor and predecessor. When a tree is balanced there would be an internal rebalancing operation, so the answer on what operations need to be modified depends on whether you want to interpret "operation" with respect to the external or the internal interface.
The answer, generally is affirmative - you'd need to keep a pointer to the successor (the next element greater than the current one) and the predecessor.
What the question is really about though, is how do you keep these newly proposed pointers up to date given insertions, deletions and rebalance. You can reason about this by induction - given a balanced tree
T such that the conditions given in the question apply with respect to
T, how inserting a new node
V will affect all the other nodes of the tree, and especially those for whom
V will become the new successor and predecessor. You will have to show that only
O(log n) nodes will be affected, located and modified, otherwise you cannot keep the operations discussed within this scope of the necessary complexity.