ikanaku/ikanakute

行かなく・行かないで/行かず・行かずに

If I'm not wrong, "ikazuni" is the polite form of "ikanai de", and "ikazu" is the polite form of ikanaku/ikanakute, right?

I've known this type of grammar for a while, but just the other day, I began tot think about it as I tried using it myself. Are "ikazu" and "ikazu ni" forms you would encounter mostly in formal written Japanese and novels, or can they be used in polite communication (such as business emails) too? Or would you rather use "ikanaku" for the latter?

Usually -naku as a verb in negative is used with a particle te. (I do not say うまく行かなく but I say うまく行かなくて .)

Personally I do not feel much difference in politeness among 行かなくて , 行かないで and 行かず(に), but 行かなくて sounds most casual while 行かず(に) sounds most formal.
Among these expressions, only -naide may work as a polite imperative.
JP: (どうか) 怒らないで [okoranaide] (ください)。
EN: Please don’t get angry.

I use -zuni when the verb in negative is combined with a following verb.
JP: わたしは 運よく [un-yoku]、けがを せずに すんだ。
EN: Luckily I did not get injured in the end.
: Here しなくて / しないで can replace せずに .
JP: わたしは 泣きさけばずに(は) いられなかった。
EN: I could not help crying.
: Because -zuni(wa) -nai is a set phrase, here 泣きさけばずに(は) cannot be replaced by -なくて / -ないで .
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When -naku is an adjective in negative, we use it with or without a particle te.
Adjectives are not used in -naide.
An adjective takes -ku form in -naku but takes -kara from in -zu.
The latter -karazu is a traditional expression but is somtimes used now especially in written Japanese.
JP: 彼が 帰ってくる 日は 遠くない [tohkunai] 。 = 彼は 遠からず [tohkarazu] 帰ってくる。
EN: It will not be long until (the day) he returns.