I did and got the "World Warrior" achievement today and I can confirm the ennemies displayed during the ending are not enough to get the achievement, those ennemies are just the ennemies of the first eight stages (so the boss stages)
I can also confirm that you can save (so, yeah, you don't need to do in straight, but you have to do it on the same save).
Here's how I did :
I started a "New game", and started with "Galaxy Man" stage. Every ennemy in this stage can be killed with the Mega Buster, and even the boss is really easy with it, then you gain the "Black Hole Bomb", so with that you'll be able to kill near all ennemies in the other stages.
Then, you can take the order you want, as the Black Hole Bomb will help you kill ennemies you wouldn't be able to kill without some other weapons.
About the boss fight, like explained in the thread, I assured myself to kill at least one ennemy for each (fish, hornet), and I even used the Black Hole Bomb on "Plug Man's shots" but I don't think it's required (I just wanted to be sure, and so to not have to do it again just because of that).
I killed the most ennemies I could anyway (and when I was losing a life, I killed the ennemies in front of me again, even if I had killed them before).
About the ballon, I just killed them by shooting the face.
In "Hornet Man", I destroyed at least one shot of each flower pots and killed each flower pot.
In Wily Stages, same, killed one ennemi of each.
The only ennemy which can't be killed is the big green thing which shoot when you shoot him.
I wouldn't advice people to use Tornado Blow because I don't see it kill the ennemies.
When ennemies are killed, you sometimes get a item (that's the case with the Black Hole Bomb but not with the Tornado Blow). You can use it when needed, but not to kill ennemies I think.
Another thing, last form of Wily shots are not needed (so no need to "Black Hole Bomb" them).