Everyone guessing there's something lost in translation are exactly right. The Japanese have a saying that if someone is tough then they "wouldn't die, even if you killed them". It doesn't literally make sense. This is because it's something called an "idiom". Pretty much every language has tons of them, people just don't notice because they're used to them. Like how you didn't question how phrases can have weight even though I said "tons of them" in the previous sentence.

In the first screencap the character is making the claim that this common Japanese idiom is, of course, literally incorrect. I think he was in some sort of life-or-death situation and was just pointing out the stakes were permanent despite the common cliche. It's actually a very reasonable line of dialogue, so it's a shame it gets laughed at out-of-context with "lol them crazy Japs!" etc.

Oh, and I think the first version of this image only had a few legit uses of "die when he/she is killed" which made more sense, but then with this version it got "beefed up" with similar lines that don't really fit - for instances of characters who fake their death, or have been resurrected, and I think "Shoot her to death, without killing her!" is just the character getting flustered and misspeaking.