है/हो/हैं /हूँ can usually be omitted in a negative sentence with नहीं. It is not wrong to include it but it is optional.
The only exception to this is if the sentence does not contain any other verb in which case है/हो/हैं /हूँ cannot be omitted. Eg: तुम बीमार हो - You are sick
Is this only the case for Simple present tense, or with other present tenses too? And what about past/future tenses? Like, could I say नेहा मछली खा नहीं रही or नेहा मछली खा नहीं चुकी?
Both your sentences are right. So, you can omit the है everywhere. Past and future tenses use the था or होगा which you can't omit (otherwise they would be the same as the present tense form without the है)