Getting the sorted sequence from a level-wise sorted min-heap
Use a recursive algorithm.  Suppose the heap has depth $D$.  Recursively compute a sorted list of all the numbers in the top $D-1$ levels (i.e., all but the bottom level); let $L$ be this list.  Let $L'$ be the list of numbers in the bottom level.  Note that both $L$ and $L'$ are in sorted order, so we can use the Merge procedure from Mergesort to compute a sorted list of all of the numbers.

If you work out the recurrence relation, you'll see that the running time of this algorithm is $O(n)$.  Indeed, Merge runs in $cn$ time, for some constant $c$, so the total running time will be $cn + cn/2 + cn/4 + \dots \le 2cn = O(n)$.