What if I would say "I do not hate spiders". "Nie nienawidzę"? And this is already genitive, because of the "nienawidzę", but the negation is negated in this case, does it still require genitive for "spiders"? Does every verb that starts with "nie-" requires genitive after them?
Yes, "nie nienawidzę", although recently I've seen somewhere here a comment from someone claiming that we don't talk like that. This really sounds fine to me, it's just not something that you need to say often.
Generally "negated Accusative changes into Genitive", and none other case changes, so negated Genitive is still Genitive.
Not sure if there are any other verbs that start with "nie", at least I can't come up with any quickly at this hour, but anyway, such a start can simply be a coincidence, so perhaps it's better not to assume too much ;)
The genitive case requirement is not obvious in these other dictionaries, as well ... www.duolingo.com/dictionary/Polish/nienawidzi%C4%87/7345cf76cb6a0736191a0bf8d5bd0590
https://pl.wiktionary.org/wiki/nienawidzi%C4%87 (viewed with Google Translate)
The Polish wiktionary says:
(1.1) nienawidzić + D.
The "D." stands for 'Dopełniacz' (genitive case).
If you click on 'składnia' you will see two options for this verb:
+ czego (the question word for the genitive case)
+ bezokolicznik (an infinitive).