Is there an equivalent of van Emde Boas trees for ropes?
Fredman and Saks show in "The cell probe complexity of dynamic data structures" that you cannot do better than $\Theta\left(\frac{\lg m}{\lg \lg m}\right)$ amortized time for the operations you are looking for. They call this problem "list representation".

Dietz presented a data structure in "Optimal Algorithms for List Indexing
and Subset Rank" that achieves this bound.