1. "Neither . . . nor" is correct. "Not . . . nor" is incorrect. In terms of the next sentences, I'm not sure whether "never . . . nor" is correct (it's usually "neither . . . nor") but in everyday conversation people rarely use "nor." It sounds a bit pretentious or literary, depending on the context. 2. Not 100% about the rule in this case, but "either . . . or" sounds better here, when you're starting with a negative.

1. "On" is correct in both cases. 2. Either sounds OK, but probably the "police's database" is more correct. 3. "Such a high expenditure" is correct because it's singular. You would say "such high expenditures" (with an s) because plural. "Badly hurt" or "do serious damage to" are both ok, but "hurt badly" sounds a little strange.

I think it's basically the same -- more a stylistic choice. There may be subtle differences but they are of no importance.

I think "give to" is correct. "Give for" means you are giving it in place of someone else -- they can't give it and you are giving it instead (for them).

"Birthers" are those who believed, despite no evidence for the idea, that Obama was not born in the United States and therefore not a citizen. (Obama made his birth certificate public.) They are people who have no respect for facts. Trump was one and I guess Arpaio was as well.