Order of steps in Kosaraju-Sharir
Yes, you are understanding this correctly.
The set of SCCs of a graph is uniquely determined. So if the algorithms were returning different sets, this would imply that one of them is incorrect. For the Wikipedia version a correctness proof is linked at the bottom of the article. If Sedgewick also gives a correctness proof of his algorithm, these two proofs together imply the property.
Reversing a graph twice will give you the original graph. Thus, if one algorithm performs especially well/bad on a certain graph, the other one will show that performance on the reversal of the graph in question. So neither algorithm will perform better in the general case.  


The comment on SO that the WP version does use $O(n)$ extra space for the stack is not entirely correct: Sedgewick also has to store the order of traversal in some way, which wll also take $O(n)$ space (though possibly with a smaller hidden constant). Additionally, you can replace the stack with a different way of storing the order in the WP algorithm as well.