A treap (or any balanced binary search tree) will be enough.

First, find out all lengths of ropes that would appear during process (for example, keep sets of cut points and lengths of ropes). After that, you can use coordinate compression and segment-sum, point-update segment tree that keeps number of ropes of given length. For "cut" queries just do three update queries, for "k-th by length" do tree descend to find smallest length l that there are at least k ropes of length l or less.