なかれ is an archaic way of speaking, found in texts such as prophecies, translations of Shakespeare, fairy tales, and ancient spells in fantasy worlds. Unless you want to intentionally imitate these styles, you should not use it yourself. It is sufficient to understand its meaning. ～するなかれ and ～することなかれ are almost the same, but the former feels older and more orthodox to me. See also: https://japanese.stackexchange.com/q/53495/5010

～ず is a common expression in modern standard Japanese. あえて誤解をおそれず can be used without any issues in a normal formal text. See: https://japanese.stackexchange.com/q/235/5010 However, in classical Japanese, ず was also used as the predicative form (終止形), so you may see ず appearing at the end of an archaic sentence. Its meaning is the same as ない. Either way, it does not have a negative imperative meaning (i.e., "Don't").

あえて is simply an adverb with the same meaning as English "dare". It means something like "doing even though it's unnecessary" or "doing even after considering the risks". It is not something to be paired with なかれ or ず for memorization. あえて～するな corresponds to "Don't dare ～" or "Don't do ～ unnecessarily" in English. As an exception, if you're doing [漢文訓読][1], あえて～ず is said to [have a special meaning][2], but you don't need to worry about it unless you're studying *kanbun* seriously. Most Japanese people learn about this once in high school, but they usually forget it by the time they graduate.

TLDR: Unless you're seriously learning old materials, focus on memorizing the usage of あえて and ず individually. Don't try to use なかれ unless you're doing some Shakespearean roleplay.


  [1]: https://en.wikipedia.org/wiki/Kanbun
  [2]: https://www.try-it.jp/chapters-14482/sections-14486/lessons-14901/practice-3/