Prove that for a general data structure - operations Extract_min() and Insert(x) cost $\Omega(\log n)$?
I don't think your proof is valid, because it only considers trees, and a certain type of trees at that. If there were an algorithm with a smaller lower bound for what you describe, we'd have a sorting algorithm faster than $\Omega(n \log n)$ no matter which of the two operations is $\log n$. So the problem reduces to proving that sorting cannot be faster than $\Omega(n \log n)$, which is a classical proof that you can find online.