Your 2nd and 3rd inquiries are correct, but as for your first and fourth they depend on what you mean by "initiative." The "initiative" you have to interrupt or negate an opponent's enhancement that is causing you to lose the battle is different than regular initiative by the numbers. (I typically don't call the "special initiative" initiative to avoid confusion.)
I think what he is trying to say is that there are two kinds of initiative, the first type, when you are losing the battle by numbers, the second type when you are losing the battle by removal (I.E. being discarded). You can only play (normal) enhancements when you have either of these types of initiative.
An example of where you wouldn't be able to play a negate / interrupt is...
You make a Rescue Attempt with Joseph, 9/7 white hero, I block with Astrologers 3/3 pale green Evil Character.
I am losing by the numbers, so I can play an enhancement.
I play Confusion 0/0, You may search opponents draw pile for any card and discard it
I am still losing by the numbers, and you aren't being removed from the battle, so you wouldn't be able to negate / interrupt my enhancement at this time.
