If the ないで clause expresses something that one is expected to do, it will be translated as “without doing ~”, but if not, it will be translated as “do not do something and ~”.
ないで is often used to combine two sentences, but it can also be used with the auxiliaries ください, 欲しい, もらう and おく.
ないで is only used with verbs. It is never used with adjectives.
If ないで is used without the auxiliaries, it will imply that the speaker is emphasizing negation, because the contents of the ないで clause are contrary to the speaker’s expectation.
If the で in ないで means “and”, then ずに can be used to replace ないで. Thus, ずに can be used in sentences 1, 2, and 3, but not in sentences 4, 5 and 6.
The difference between ないで and ずに is that ないで is used in informal speech or conversational Japanese, whereas ずに is used in formal speech or written Japanese.
ないで can be replaced by なくて only when the ないで clause indicates some cause for emotion which is expressed in the main clause. ずに cannot be used in such sentences.
僕は英語が話せないで恥ずかしかった。 ― boku wa eigo ga hanasenaide hazukashikatta. ― I felt ashamed because I couldn’t speak English.
僕は英語が話せなくて恥ずかしかった。 ― boku wa eigo ga hanasenakute hazukashikatta. ― I felt ashamed because I couldn’t speak English.
Sentences that are structured as “X is not Y but X” use なくて instead of ないで.
私は学生ではなくて先生です。 ― watashi wa gakusei de wa nakute sensei desu. ― I am not a student but a teacher.[1]
^ Makino, Seiichi, Tsutsui, Michio (1989 January 1) “Main Entries: -nai de”, in A Dictionary of Basic Japanese Grammar, 1st edition, 5-4, Shibaura 4-chome, Minato-ku, Tokyo 108-0023, Japan: The Japan Times, →ISBN, pages 271-273