Is there an O(1) solution to find the kth-smallest element in an implicit min-heap?
This is not possible. If this is the case then we could extract a sorted list in linear time; $O(1)$ for each $k$th smallest element. We can also create a min heap in linear time thus we could sort in linear time! We know comparison sorting has an $\Omega(n \log n)$ lower bound worst case, so this is not possible.