Learn Russian: Здесь ни звука, ни намёка на жизнь. - There is not a sound here, not a hint of life.

This sentence uses verbless existential negation. In Russian, you can express the absolute absence of something by simply using the particle ни followed by a noun in the genitive case. You do not need a verb or even the word нет; the structure ни + genitive inherently means that not a single instance of the noun exists in that location.

Could I use нет instead, such as Здесь нет звука, нет намёка на жизнь?

Yes, but the nuance would change. Using нет simply states the factual absence of those things. Using the repeated ни ... ни ... structure without нет is highly emphatic and slightly more literary. It translates closer to 'Not a single sound, not a single hint', strongly emphasizing the absolute emptiness of the place.

Why does 'of life' translate to на жизнь using the accusative case instead of just the genitive жизни?

This is dictated by the noun намёк (hint). In Russian, when you talk about a hint or allusion to something, you must use the preposition на followed by the accusative case. Therefore, 'a hint of life' is always намёк на жизнь, just like 'a hint at the truth' would be намёк на правду.

How would I say this in the past tense if there is no verb to change?