Is this lower bound proof for the comparison-based sorting problem correct?
No, I don't think it's a reasonable proof.  This makes claims without justification.  For instance, the second bullet says that the only way to put an element into its final position is (the algorithm must know something..).  How do you know that is correct?  How would you prove that?  What does it even mean for an algorithm to know something?  Why does knowing that cost at least $\Omega(\lg i)$ comparisons?  All of that seems a bit unclear to me, so I don't find it a persuasive proof.