Assault Suit Leynos

This trophy can also be a pain in the butt. I've been able to do it on stage 1 and stage 5.

For stage 1, proceed slowly and carefully, taking out the enemies one at a time and making sure to block or avoid their shots. When the boss ship appears, stay in front of it, but not too far in front, and walk backwards, alternating between shooting it with the D-D and machine gun. I suggest unequipping the 25-SC to save time on weapon swaps. Be ready to shoot any smaller enemies that catch up to you, and block their shots with the shield. When the boss is about to shoot the long stream of red bullets, use the booster to quickly retreat to the left and wait until the barrage ceases, then go back and continue shooting until you take him down.

For stage 5, use the same method outlined for getting "Bulkhead Protector": Missiles and LG-GUN / HG-GUN. Definitely don't equip armor as that would be a waste of a slot since you're going for no damage taken at all. And you'll absolutely need to equip the shield. Most of the enemies don't shoot at you, so just take care not to be hit by the bombs the planes drop. After clearing the last wave, make sure you have missiles ready (M.POD works best) and hold down so you'll have the shield raised as soon as Lance appears. Block his flame shots and hit him with everything you've got to take him out quickly. He does fire a few missiles as well, so be ready to block, dodge or shoot them down with your own missiles.

On stage 2 you will be attacked from both sides, and stage 3 will have enemies and bullets flying at you constantly, so I do not recommend trying for this trophy on either of those stages.