A. Rotation in AVL trees , keeps the AVL tree still a binary search tree , thus the inorder numbering doesn't change .
B. Let's come to the second statement , the description given is of a full binary tree.
Now if we imagine a full binary tree ,
In binary search tree , we know that time to search is $O(h)$ , where $h$ is the height of the binary tree.
Let's calculate the height in this case of full binary tree,
Total number of nodes = $n = 1+2+2+2+.... h$ times (2 nodes at each level) = $1+2h$.
$1+2h=n => h=\frac{n-1}{2}$ .
Thus in this case , time to search = $O(h) = O(\frac{n-1}{2}) = O(n).$
Thus the statement is false.