ずにWithout doing

ずに has the core meaning of 'without doing', while ずにはいられない is an idiomatic form meaning 'I can't help but (A)'. Compare '宿題をせずに寝た' ('I went to bed without doing my homework'), which describes inaction, to '彼の話に笑わずにはいられなかった' ('I couldn't help but laugh at his story'), which expresses an uncontrollable response.

ずに has the core meaning of 'without doing', and ずに済む is an idiomatic usage of meaning 'not having to do (A)' (literally, 'it works out without doing it'). Compare 宿題をせずに寝た ('I went to bed without doing my homework.'), which simply describes lack of action, to e.g. 新しいスマホを買わずに済んだ ('It worked out so that I didn't have to buy a new smartphone.').

Both ずに and ことなく mean 'without doing (A)' and can often be used interchangeably. ずに is more common and used in both writing and conversation, e.g. '宿題をせずに寝た' ('I went to bed without doing my homework'). ことなく is more formal and tends to appear in writing, e.g. '彼は休むことなく働き続けた' ('He continued working without rest').

ないで and ずに both express the idea of 'without doing (A)' and are used in similar ways. For example, 朝ご飯を食べないで学校に行った and 朝ご飯を食べずに学校に行った both mean 'I went to school without eating breakfast.', but ずに has a slightly more formal or literary tone, while ないで is more conversational. Unlike ないで, ずに cannot be used as a negative request form to say 'Don't (A).'.