～ず and ～ない mean essentially the same thing, as they are both negative forms (i.e. they both mean "not"). ず is more of a written or formal style, while ない is spoken or standard.
However, the examples in your question actually revolve around ～に and ～と, as it's a grammar usage question. Let's take a look at your examples:
座【すわ】らずに is the correct answer here, and has several close analogues that could be used as well:
The nuances may vary slightly between these, but they all fit in the sentence grammatically, which is what these questions are designed to test.
In this example, the time (5月に) is the hint and the key is the と after the blank. Though ～ず can be followed by と, it ends up having the same meaning as ～ずに, so the answer has to be ならない, as ～ないと deals with time.
Though the meanings of ～ず and ～ない are basically the same, the meanings diverge when they become ～ずに and ～ないと.
- ～ずに means basically "without doing ～" (～に just turns this into an adverb, just like the で in ～ないで)
- ～ないと on the other hand, means "when not doing ～" or "if [it] doesn't ～" (～と in this case means "when" or "if")