Finding the $k$th largest element in an evolving query data structure
Have a look at the Wikipedia page for segment trees:


  It is, in principle, a static structure; that is, its content cannot be modified once the structure is built.


So it does not seem to be a good data structure for your scenario at all.

Note that the kind of query you want to perform is called selection (unimaginative, I know). Check the linked article for loads of algorithms. One simple way to solve your problem is this:


Store elements in a sorted list. That means, updates (that is insertions) take linear time.
For the queries, run a selection algorithm on the specified interval.


This approach can be improved by extending the list to a skip list.

Another approach is using binary search trees. If you use a self-balancing variant, that makes updates possible in $O(\log n)$ time. The queries you want can be implemented by storing in every node how many descendants it has, so we can find the $k$-th largest element in $O(\log n)$ time.