I'm Canadian, and when I look at your two examples, I would use "nor". Not to sound proper or scholarly, but just because "it shouldn't anyway" doesn't sound right. It's the 'anyway' part that doesn't make sense there. To me, your example sounds like someone wrote such a poorly written essay to apply for a program in a university, that it won't get them into a good university, and shouldn't because of its content. Let's use another example aside from universities. Let's say a man is using a dating app. On his profile, he lies about his success and lies about not having children. His goal is to attract as many women as possible. But a friend tells him, "A good woman will not settle for a man who lies, nor should she." What this means is that the friend is telling him that he not only condemns his lie, but believes that women are in the right to feel the same. Tasha has a good answer, but it doesn't necessarily mean "it's not important". To me, the "nor should it" part means it's obvious and it should be taken into consideration that it's a problem no matter what. "This won't get you into a good university, nor should it." It just sounds like no respecting university should be settling on an application that isn't written well, for example, no matter who it is.

US English - The first one. The second isn't wrong, but it sounds weird.

Nor is more proper, and more like British English. Most Americans don't use the word "nor" very often. It's more of an academic word for more educated people. Either is correct.

Jaaguar and Tasha are correct: the accepted word order is "nor should it".

No, you mustn't switch the word order to "nor it should". "Nor should it" is correct.