"Neither" plural agreement [duplicate]

This is a common dilemma, to which, for once, there is a definite answer. There is ,in addition, a third such term: each.

A) The terms (neither and each) are singular and so take a singular verb: the verb/attribute applies to each singly.

B) The term both is plural and so takes a plural verb: the verb/attribute applies to both plurally.

BUT There is an understandable mental urge to let proximity rather than strict grammatical number determine the grammatical number of the verb/attribute rather than the number of the strict grammatical subject.

So in your examples 1 and 2, the strict grammatical subject is singular and so the verb should follow suit: has.

But in 2 the plural is often have, triggered by the adjacent plural noun things, even though it is not the strict subject. Is it an error? In my own schooldays I would say yes, but this plural shift is now too common to render criticism churlish. If your audience is likely to judge you or what you write adversely if it violates the strict rule, stick to it. Otherwise don't worry. Personally I wince to see "neither of these pupils have passed the test.", but, in the words of Clytemnestra, "an ox sits on my tongue". It is common enough to count as established.