Other have explained what "it" refers to, though I think it's closer to an impersonal it. I had to read these sentences several times to understand the sense of #3, it's sloppy. And even now that I do, it doesn't match the sense of #4. No, "prevent" cannot mean "help" here. On its own, #3 could be plainly restated: "Even if *we have to work more hours and less safely*, it (meaning the need for drastic measures) won't stop us from succeeding." It's hard to understand, but since there's a lot of odd and obscure phrases here, I guess it's a piece of dialogue which emphasizes the odd way the person speaks (writing "gonna" is also something you would only do when writing dialogue or communicating very informally). Also, a "quixotic" goal is one that is essentially unobtainable--one that makes sense only in fantasy. It's a weird adjective to apply to any goal you actually plan to achieve. In #1, the second "sentence" is not a sentence, and I can't guess what it means. Likewise, I can't guess what the second sentence in #2 means. Google results for "Lickety-quick" are dominated by a specific educational game, but otherwise it's very obscure. It sounds more like a mistake for "lickety-split" than anything else. But the reason I can't figure out what it means is that I don't know what "shots" means in this context. Sentence #4 makes sense, but not in combination with #3. "The best we can hope for" only makes sense if you assume the quota won't be met. I note that substituting "missing" for "meeting" in #3 makes both make sense (and #3 much clearer). A "round" in this context would probably refer to something specific--a round of ammunition, or some other product that comes in "rounds", like cakes or pieces of wood. If it's referring to collective units of a factory's production (for example), "batches" is probably better; if it's referring to the number of times a factor produces output, "runs" is probably better.

In the third sentence, "it" refers to the whole preceding idea, "Even if we're double-timing it around the clock with nary a carey for workplace safety, ...". Sometimes, it is a good idea to use "this" instead of "it" in such cases, to avoid ambiguity. But "it" is correct too. By the way, "nary a carey" is a humorous (and very unusual) equivalent of "hardly a care" (= almost not worrying about workplace safety at all").

[it] refers to the work speed, or work efficiency of that group of workers.