With regard to your first question, I claim that global choice is equivalent to the assertion that any two classes are comparable under injectivity.
If global choice holds, then this is clearly the case. Conversely, consider the classes Ord and the class $W$ that I used in my answer to your previous question. If Ord injects into $W$, then global choice holds, since that is what I had argued there. And if $W$ injects into Ord, then we can well-order $W$ and from this we may construct a well-ordering of $V$.
Thus, if global choice fails, there must be incomparable classes.