Fibonacci heaps have $O(1)$ insertion and $O(\log n)$ delete-min and delete-key (under amortized complexity). Is there a heap data structure with $O(1)$ insertion and delete-key and $O(\log n)$ delete-min and find-min? That is, can delete-key be reduced to $O(1)$ at the cost of more expensive other operations such as find-min, meld, and reduce-key? Here delete-key means remove a particular key given a pointer to that key in the data structure.
This data structure would be necessary to fix my broken answer to this question: Nontrivial algorithm for computing a sliding window median.