Data structure with insert, and delete-min (or max) in $O(1)$?
No; at least, it seems unlikely.  Such a data structure would contradict the lower bound on comparison-based sorting algorithms and would imply linear-time sorting algorithms (where none is known to exist).

Suppose we want to sort $m$ numbers from the range $1..n$, where $m \ll n$.  If we had your data structure, we could do that in $O(m)$ time.  However, this seems implausible.  The fastest comparison-based algorithm runs in time $\Theta(m \log m)$, which is  larger than $O(m)$.  Counting sort takes $O(m+n)$ time, which is much larger than $O(m)$ when $m \ll n$.  Van Emde Boas sorting takes $O(m \log \log n)$ time, which is still larger than $O(m)$ time. Han &amp; Thorup's sorting algorithm takes $O(m \sqrt{\log \log n})$ time, which is still larger than $O(m)$ time.

So, you should not expect there to be any data structure of the form you list, as that would imply a huge breakthrough in sorting algorithms.

However, there is probably an algorithm where those operations can be done in $O(\log \log n)$ time, or possibly even $O(\sqrt{\log \log n})$ time, by applying Van Emde Boas trees or Han &amp; Thorup.  See also https://en.wikipedia.org/wiki/Integer_sorting#Sorting_versus_integer_priority_queues.