Asymptotic bounds of the following operations
If you could do all three operations faster than $O(\log n)$ time, then you would obtain a sorting algorithm whose running time is faster than $O(n \log n)$ time.  That's known to be impossible for comparison-based algorithms.
So, you probably can't do better than $O(\log n)$ time for all three operations.  In particular, you definitely can't do better using a comparison-based algorithm (one that only compares items but has no other knowledge about the values of items).
The reduction works as follows.  First, scan through the list of $n$ numbers and insert each into your data structure.  Second, scan through the list again; for each number $x$, look up its index, and then write it into an output array at that index.  This does $2n$ operations on your data structure, so if your data structure can support insert and lookup in $O(\log n)$ time, the total running time will be $O(n \log n)$.